GCSE Maths: Factorising quadratic expressions (Higher)
Part of the GCSE Maths quadratic and simultaneous equations chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Factorising quadratic expressions
In Chapter 7 you expanded brackets: (x + 2)(x + 3) = x2 + 5x + 6. Factorising is the same journey backwards. You start with x2 + 5x + 6 and look for the brackets that multiply to make it. You need this skill to solve quadratic equations, so it is worth becoming really fluent at it before you move on.
A quadratic expression is an expression whose highest power is 2, such as x2 + 5x + 6. A factor is something that is multiplied: (x + 2) and (x + 3) are factors of x2 + 5x + 6.
Look at the picture. The whole rectangle has area x2 + 3x + 2x + 6 = x2 + 5x + 6, and its sides are (x + 2) and (x + 3). Now look at the numbers: 2 × 3 = 6, which is the last number, and 2 + 3 = 5, which is the middle number (the number in front of x). This happens every time, and it gives us a method.
To factorise x2 + bx + c: find two numbers that multiply to make c and add to make b. Call them p and q. Then the answer is (x + p)(x + q). The signs follow a pattern. If c is positive, both numbers have the same sign as b. If c is negative, one number is positive and one is negative, and the bigger one has the sign of b.
Always look for a common factor first. In 3x2 − 12x every term divides by 3x, so it factorises as 3x(x − 4). Check by expanding.
Difference of two squares. An expression with two terms, both perfect squares, with a minus between them, factorises as a2 − b2 = (a − b)(a + b). Why? Expand (a − b)(a + b) and the two middle terms, +ab and −ab, cancel out. So x2 − 49 = (x − 7)(x + 7). There is no middle term because there is no x term.
Quadratics where the number in front of x2 is not 1 (like 2x2 + 7x + 3) need one more step, called splitting the middle term. Multiply the first and last numbers (2 × 3 = 6). Find two numbers that multiply to make that (6) and add to make the middle number (7): they are 1 and 6. Split 7x into 1x + 6x, then factorise the front pair and the back pair separately: 2x2 + x + 6x + 3 = x(2x + 1) + 3(2x + 1) = (2x + 1)(x + 3). The matching brackets show it has worked.
Key ideas
- Factorising is the reverse of expanding. Always check by expanding your answer.
- x2 + bx + c = (x + p)(x + q) where p × q = c and p + q = b.
- Take out a common factor first: 3x2 − 12x = 3x(x − 4).
- Difference of two squares: a2 − b2 = (a − b)(a + b).
- Higher: for ax2 + bx + c, find two numbers that multiply to ac and add to b, split the middle term, and factorise in pairs.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares (two square terms with a minus between)? (3) is it a quadratic with three terms?x2 + 7x + 12
- Factorise. Follow the method for the pattern you found.Spot the pattern. This is a quadratic of the form x² + bx + c, with b = 7 and c = 12.
We need two numbers that multiply to 12 (the last number) and add to 7 (the middle number). List the pairs that multiply to 12:
1 × 12 = 12, add: 1 + 12 = 13 ✗
-1 × -12 = 12, add: -1 + -12 = -13 ✗
2 × 6 = 12, add: 2 + 6 = 8 ✗
-2 × -6 = 12, add: -2 + -6 = -8 ✗
3 × 4 = 12, add: 3 + 4 = 7 ✓ this pair works
Put the pair into two brackets: (x + 3)(x + 4).
Check by expanding: x² + 4x + 3x + 12 = x2 + 7x + 12. ✓
x2 + 7x + 12 = (x + 3)(x + 4) - Check by expanding. Factorising is the reverse of expanding, so multiplying the brackets out must give the original expression back.First: x × x = x2
Outer: x × 4 = 4x
Inner: 3 × x = 3x
Last: 3 × 4 = 12
(x + 3)(x + 4) = x2 + 7x + 12 ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares (two square terms with a minus between)? (3) is it a quadratic with three terms?x2 − 2x − 15
- Factorise. Follow the method for the pattern you found.Spot the pattern. This is a quadratic of the form x² + bx + c, with b = -2 and c = -15.
We need two numbers that multiply to -15 (the last number) and add to -2 (the middle number). List the pairs that multiply to -15:
1 × -15 = -15, add: 1 + -15 = -14 ✗
-1 × 15 = -15, add: -1 + 15 = 14 ✗
3 × -5 = -15, add: 3 + -5 = -2 ✓ this pair works
Put the pair into two brackets: (x + 3)(x − 5).
Check by expanding: x² − 5x + 3x − 15 = x2 − 2x − 15. ✓
x2 − 2x − 15 = (x + 3)(x − 5) - Check by expanding. Factorising is the reverse of expanding, so multiplying the brackets out must give the original expression back.First: x × x = x2
Outer: x × 3 = 3x
Inner: (−5) × x = −5x
Last: (−5) × 3 = −15
(x + 3)(x − 5) = x2 − 2x − 15 ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares (two square terms with a minus between)? (3) is it a quadratic with three terms?x2 − 49
- Factorise. Follow the method for the pattern you found.Spot the pattern. There are two terms, both are perfect squares, and there is a minus between them. This is a difference of two squares.
The rule is: a² − b² = (a − b)(a + b).
Find a and b. x2 = (x)², so a = x. And 49 = 7², so b = 7.
Put them into the rule: (x − 7)(x + 7).
Check by expanding: (x − 7)(x + 7) = x2 + 7x − 7x − 49 = x2 − 49. ✓ (the middle terms cancel)
x2 − 49 = (x − 7)(x + 7) - Check by expanding. Factorising is the reverse of expanding, so multiplying the brackets out must give the original expression back.First: x × x = x2
Outer: x × 7 = 7x
Inner: (−7) × x = −7x
Last: (−7) × 7 = −49
(x − 7)(x + 7) = x2 − 49 ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares (two square terms with a minus between)? (3) is it a quadratic with three terms?5x2 − 20
- Factorise. Follow the method for the pattern you found.Look for a common factor. A common factor is something that divides into every term.
The terms are 5x2, −20. The biggest number and the highest power of x that divide into all of them is 5.
Take it outside a bracket, and inside the bracket put each term divided by it:
5x2 ÷ 5 = x2
−20 ÷ 5 = −4
So 5x2 − 20 = 5(x2 − 4).
Check by expanding: 5 × (x2 − 4) = 5 × x2, 5 × (−4) → 5x2 − 20. ✓
Can the bracket be factorised again? Look at (x2 − 4).
Spot the pattern. There are two terms, both are perfect squares, and there is a minus between them. This is a difference of two squares.
The rule is: a² − b² = (a − b)(a + b).
Find a and b. x2 = (x)², so a = x. And 4 = 2², so b = 2.
Put them into the rule: (x − 2)(x + 2).
Check by expanding: (x − 2)(x + 2) = x2 + 2x − 2x − 4 = x2 − 4. ✓ (the middle terms cancel)
5x2 − 20 = 5(x − 2)(x + 2) - Check by expanding. Factorising is the reverse of expanding, so multiplying the brackets out must give the original expression back.First: 5x × x = 5x2
Outer: 5x × 2 = 10x
Inner: (−10) × x = −10x
Last: (−10) × 2 = −20
5(x − 2)(x + 2) = 5x2 − 20 ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares (two square terms with a minus between)? (3) is it a quadratic with three terms?2x2 + 7x + 3
- Factorise. Follow the method for the pattern you found.Spot the pattern. A quadratic where the number in front of x² is 2, not 1. We use the split the middle term method.
Multiply the first and last numbers: 2 × (3) = 6. We need two numbers that multiply to 6 and add to 7 (the middle number).
1 × 6 = 6, add: 7 ✓ this pair works
Split the middle term (7x) into 1x and 6x: 2x2 + 1x + 6x + 3.
Factorise each pair separately. First pair: 2x2 + x = x(2x + 1). Second pair: 6x + 3 = 3(2x + 1).
The two brackets (2x + 1) are the same. That is how we know it has worked. Write: x(2x + 1) + 3(2x + 1).
Take the common bracket out: (2x + 1) multiplied by (x + 3).
Result: (x + 3)(2x + 1).
Check by expanding: (x + 3)(2x + 1) = 2x2 + 7x + 3. ✓
2x2 + 7x + 3 = (x + 3)(2x + 1) - Check by expanding. Factorising is the reverse of expanding, so multiplying the brackets out must give the original expression back.First: x × 2x = 2x2
Outer: x × 1 = x
Inner: 3 × 2x = 6x
Last: 3 × 1 = 3
(x + 3)(2x + 1) = 2x2 + 7x + 3 ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares (two square terms with a minus between)? (3) is it a quadratic with three terms?6x2 − x − 2
- Factorise. Follow the method for the pattern you found.Spot the pattern. A quadratic where the number in front of x² is 6, not 1. We use the split the middle term method.
Multiply the first and last numbers: 6 × (-2) = -12. We need two numbers that multiply to -12 and add to -1 (the middle number).
1 × -12 = -12, add: -11 ✗
-1 × 12 = -12, add: 11 ✗
2 × -6 = -12, add: -4 ✗
-2 × 6 = -12, add: 4 ✗
3 × -4 = -12, add: -1 ✓ this pair works
Split the middle term (-1x) into 3x and -4x: 6x2 + 3x − 4x − 2.
Factorise each pair separately. First pair: 6x2 + 3x = 3x(2x + 1). Second pair: −4x − 2 = −2(2x + 1).
The two brackets (2x + 1) are the same. That is how we know it has worked. Write: 3x(2x + 1) − 2(2x + 1).
Take the common bracket out: (2x + 1) multiplied by (3x − 2).
Result: (2x + 1)(3x − 2).
Check by expanding: (2x + 1)(3x − 2) = 6x2 − x − 2. ✓
6x2 − x − 2 = (2x + 1)(3x − 2) - Check by expanding. Factorising is the reverse of expanding, so multiplying the brackets out must give the original expression back.First: 2x × 3x = 6x2
Outer: 2x × (−2) = −4x
Inner: 1 × 3x = 3x
Last: 1 × (−2) = −2
(2x + 1)(3x − 2) = 6x2 − x − 2 ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Watch out
The expression x2 + 9 cannot be factorised, because there is a plus between the squares. Only a minus gives a difference of two squares. Also, do not forget to take out a common factor first: 2x2 − 50 is 2(x − 5)(x + 5), not just 2(x2 − 25).Exercise 14A Factorising quadratic expressions
Non-calculatorKey words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?6x + 9
- Factorise. Follow the method.Look for a common factor. A common factor is something that divides into every term.
The terms are 6x, 9. The biggest number and the highest power of x that divide into all of them is 3.
Take it outside a bracket, and inside the bracket put each term divided by it:
6x ÷ 3 = 2x
9 ÷ 3 = 3
So 6x + 9 = 3(2x + 3).
Check by expanding: 3 × (2x + 3) = 3 × 2x, 3 × 3 → 6x + 9. ✓
Check. Expand your answer: 3(2x + 3) should give 6x + 9 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?10x − 15
- Factorise. Follow the method.Look for a common factor. A common factor is something that divides into every term.
The terms are 10x, −15. The biggest number and the highest power of x that divide into all of them is 5.
Take it outside a bracket, and inside the bracket put each term divided by it:
10x ÷ 5 = 2x
−15 ÷ 5 = −3
So 10x − 15 = 5(2x − 3).
Check by expanding: 5 × (2x − 3) = 5 × 2x, 5 × (−3) → 10x − 15. ✓
Check. Expand your answer: 5(2x − 3) should give 10x − 15 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?x2 + 5x
- Factorise. Follow the method.Look for a common factor. A common factor is something that divides into every term.
The terms are x2, 5x. The biggest number and the highest power of x that divide into all of them is x.
Take it outside a bracket, and inside the bracket put each term divided by it:
x2 ÷ x = x
5x ÷ x = 5
So x2 + 5x = x(x + 5).
Check by expanding: x × (x + 5) = x × x, x × 5 → x2 + 5x. ✓
Check. Expand your answer: x(x + 5) should give x2 + 5x again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?4x2 − 12x
- Factorise. Follow the method.Look for a common factor. A common factor is something that divides into every term.
The terms are 4x2, −12x. The biggest number and the highest power of x that divide into all of them is 4x.
Take it outside a bracket, and inside the bracket put each term divided by it:
4x2 ÷ 4x = x
−12x ÷ 4x = −3
So 4x2 − 12x = 4x(x − 3).
Check by expanding: 4x × (x − 3) = 4x × x, 4x × (−3) → 4x2 − 12x. ✓
Check. Expand your answer: 4(x − 3)x should give 4x2 − 12x again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?9x2 + 6x
- Factorise. Follow the method.Look for a common factor. A common factor is something that divides into every term.
The terms are 9x2, 6x. The biggest number and the highest power of x that divide into all of them is 3x.
Take it outside a bracket, and inside the bracket put each term divided by it:
9x2 ÷ 3x = 3x
6x ÷ 3x = 2
So 9x2 + 6x = 3x(3x + 2).
Check by expanding: 3x × (3x + 2) = 3x × 3x, 3x × 2 → 9x2 + 6x. ✓
Check. Expand your answer: 3x(3x + 2) should give 9x2 + 6x again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?8x2 − 8x
- Factorise. Follow the method.Look for a common factor. A common factor is something that divides into every term.
The terms are 8x2, −8x. The biggest number and the highest power of x that divide into all of them is 8x.
Take it outside a bracket, and inside the bracket put each term divided by it:
8x2 ÷ 8x = x
−8x ÷ 8x = −1
So 8x2 − 8x = 8x(x − 1).
Check by expanding: 8x × (x − 1) = 8x × x, 8x × (−1) → 8x2 − 8x. ✓
Check. Expand your answer: 8(x − 1)x should give 8x2 − 8x again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?x2 + 5x + 6
- Factorise. Follow the method.Spot the pattern. This is a quadratic of the form x² + bx + c, with b = 5 and c = 6.
We need two numbers that multiply to 6 (the last number) and add to 5 (the middle number). List the pairs that multiply to 6:
1 × 6 = 6, add: 1 + 6 = 7 ✗
-1 × -6 = 6, add: -1 + -6 = -7 ✗
2 × 3 = 6, add: 2 + 3 = 5 ✓ this pair works
Put the pair into two brackets: (x + 2)(x + 3).
Check by expanding: x² + 3x + 2x + 6 = x2 + 5x + 6. ✓
Check. Expand your answer: (x + 2)(x + 3) should give x2 + 5x + 6 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?x2 + 7x + 10
- Factorise. Follow the method.Spot the pattern. This is a quadratic of the form x² + bx + c, with b = 7 and c = 10.
We need two numbers that multiply to 10 (the last number) and add to 7 (the middle number). List the pairs that multiply to 10:
1 × 10 = 10, add: 1 + 10 = 11 ✗
-1 × -10 = 10, add: -1 + -10 = -11 ✗
2 × 5 = 10, add: 2 + 5 = 7 ✓ this pair works
Put the pair into two brackets: (x + 2)(x + 5).
Check by expanding: x² + 5x + 2x + 10 = x2 + 7x + 10. ✓
Check. Expand your answer: (x + 2)(x + 5) should give x2 + 7x + 10 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?x2 + 8x + 15
- Factorise. Follow the method.Spot the pattern. This is a quadratic of the form x² + bx + c, with b = 8 and c = 15.
We need two numbers that multiply to 15 (the last number) and add to 8 (the middle number). List the pairs that multiply to 15:
1 × 15 = 15, add: 1 + 15 = 16 ✗
-1 × -15 = 15, add: -1 + -15 = -16 ✗
3 × 5 = 15, add: 3 + 5 = 8 ✓ this pair works
Put the pair into two brackets: (x + 3)(x + 5).
Check by expanding: x² + 5x + 3x + 15 = x2 + 8x + 15. ✓
Check. Expand your answer: (x + 3)(x + 5) should give x2 + 8x + 15 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?x2 + 9x + 20
- Factorise. Follow the method.Spot the pattern. This is a quadratic of the form x² + bx + c, with b = 9 and c = 20.
We need two numbers that multiply to 20 (the last number) and add to 9 (the middle number). List the pairs that multiply to 20:
1 × 20 = 20, add: 1 + 20 = 21 ✗
-1 × -20 = 20, add: -1 + -20 = -21 ✗
2 × 10 = 20, add: 2 + 10 = 12 ✗
-2 × -10 = 20, add: -2 + -10 = -12 ✗
4 × 5 = 20, add: 4 + 5 = 9 ✓ this pair works
Put the pair into two brackets: (x + 4)(x + 5).
Check by expanding: x² + 5x + 4x + 20 = x2 + 9x + 20. ✓
Check. Expand your answer: (x + 4)(x + 5) should give x2 + 9x + 20 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?x2 + 6x + 8
- Factorise. Follow the method.Spot the pattern. This is a quadratic of the form x² + bx + c, with b = 6 and c = 8.
We need two numbers that multiply to 8 (the last number) and add to 6 (the middle number). List the pairs that multiply to 8:
1 × 8 = 8, add: 1 + 8 = 9 ✗
-1 × -8 = 8, add: -1 + -8 = -9 ✗
2 × 4 = 8, add: 2 + 4 = 6 ✓ this pair works
Put the pair into two brackets: (x + 2)(x + 4).
Check by expanding: x² + 4x + 2x + 8 = x2 + 6x + 8. ✓
Check. Expand your answer: (x + 2)(x + 4) should give x2 + 6x + 8 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?x2 + 10x + 24
- Factorise. Follow the method.Spot the pattern. This is a quadratic of the form x² + bx + c, with b = 10 and c = 24.
We need two numbers that multiply to 24 (the last number) and add to 10 (the middle number). List the pairs that multiply to 24:
1 × 24 = 24, add: 1 + 24 = 25 ✗
-1 × -24 = 24, add: -1 + -24 = -25 ✗
2 × 12 = 24, add: 2 + 12 = 14 ✗
-2 × -12 = 24, add: -2 + -12 = -14 ✗
3 × 8 = 24, add: 3 + 8 = 11 ✗
-3 × -8 = 24, add: -3 + -8 = -11 ✗
4 × 6 = 24, add: 4 + 6 = 10 ✓ this pair works
Put the pair into two brackets: (x + 4)(x + 6).
Check by expanding: x² + 6x + 4x + 24 = x2 + 10x + 24. ✓
Check. Expand your answer: (x + 4)(x + 6) should give x2 + 10x + 24 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?x2 − 6x + 8
- Factorise. Follow the method.Spot the pattern. This is a quadratic of the form x² + bx + c, with b = -6 and c = 8.
We need two numbers that multiply to 8 (the last number) and add to -6 (the middle number). List the pairs that multiply to 8:
1 × 8 = 8, add: 1 + 8 = 9 ✗
-1 × -8 = 8, add: -1 + -8 = -9 ✗
2 × 4 = 8, add: 2 + 4 = 6 ✗
-2 × -4 = 8, add: -2 + -4 = -6 ✓ this pair works
Put the pair into two brackets: (x − 2)(x − 4).
Check by expanding: x² − 4x − 2x + 8 = x2 − 6x + 8. ✓
Check. Expand your answer: (x − 4)(x − 2) should give x2 − 6x + 8 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?x2 − 7x + 10
- Factorise. Follow the method.Spot the pattern. This is a quadratic of the form x² + bx + c, with b = -7 and c = 10.
We need two numbers that multiply to 10 (the last number) and add to -7 (the middle number). List the pairs that multiply to 10:
1 × 10 = 10, add: 1 + 10 = 11 ✗
-1 × -10 = 10, add: -1 + -10 = -11 ✗
2 × 5 = 10, add: 2 + 5 = 7 ✗
-2 × -5 = 10, add: -2 + -5 = -7 ✓ this pair works
Put the pair into two brackets: (x − 2)(x − 5).
Check by expanding: x² − 5x − 2x + 10 = x2 − 7x + 10. ✓
Check. Expand your answer: (x − 5)(x − 2) should give x2 − 7x + 10 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?x2 − 8x + 15
- Factorise. Follow the method.Spot the pattern. This is a quadratic of the form x² + bx + c, with b = -8 and c = 15.
We need two numbers that multiply to 15 (the last number) and add to -8 (the middle number). List the pairs that multiply to 15:
1 × 15 = 15, add: 1 + 15 = 16 ✗
-1 × -15 = 15, add: -1 + -15 = -16 ✗
3 × 5 = 15, add: 3 + 5 = 8 ✗
-3 × -5 = 15, add: -3 + -5 = -8 ✓ this pair works
Put the pair into two brackets: (x − 3)(x − 5).
Check by expanding: x² − 5x − 3x + 15 = x2 − 8x + 15. ✓
Check. Expand your answer: (x − 5)(x − 3) should give x2 − 8x + 15 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?x2 − 9x + 14
- Factorise. Follow the method.Spot the pattern. This is a quadratic of the form x² + bx + c, with b = -9 and c = 14.
We need two numbers that multiply to 14 (the last number) and add to -9 (the middle number). List the pairs that multiply to 14:
1 × 14 = 14, add: 1 + 14 = 15 ✗
-1 × -14 = 14, add: -1 + -14 = -15 ✗
2 × 7 = 14, add: 2 + 7 = 9 ✗
-2 × -7 = 14, add: -2 + -7 = -9 ✓ this pair works
Put the pair into two brackets: (x − 2)(x − 7).
Check by expanding: x² − 7x − 2x + 14 = x2 − 9x + 14. ✓
Check. Expand your answer: (x − 7)(x − 2) should give x2 − 9x + 14 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?x2 − 10x + 21
- Factorise. Follow the method.Spot the pattern. This is a quadratic of the form x² + bx + c, with b = -10 and c = 21.
We need two numbers that multiply to 21 (the last number) and add to -10 (the middle number). List the pairs that multiply to 21:
1 × 21 = 21, add: 1 + 21 = 22 ✗
-1 × -21 = 21, add: -1 + -21 = -22 ✗
3 × 7 = 21, add: 3 + 7 = 10 ✗
-3 × -7 = 21, add: -3 + -7 = -10 ✓ this pair works
Put the pair into two brackets: (x − 3)(x − 7).
Check by expanding: x² − 7x − 3x + 21 = x2 − 10x + 21. ✓
Check. Expand your answer: (x − 7)(x − 3) should give x2 − 10x + 21 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?x2 − 5x + 4
- Factorise. Follow the method.Spot the pattern. This is a quadratic of the form x² + bx + c, with b = -5 and c = 4.
We need two numbers that multiply to 4 (the last number) and add to -5 (the middle number). List the pairs that multiply to 4:
1 × 4 = 4, add: 1 + 4 = 5 ✗
-1 × -4 = 4, add: -1 + -4 = -5 ✓ this pair works
Put the pair into two brackets: (x − 1)(x − 4).
Check by expanding: x² − 4x − 1x + 4 = x2 − 5x + 4. ✓
Check. Expand your answer: (x − 4)(x − 1) should give x2 − 5x + 4 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?x2 + 2x − 8
- Factorise. Follow the method.Spot the pattern. This is a quadratic of the form x² + bx + c, with b = 2 and c = -8.
We need two numbers that multiply to -8 (the last number) and add to 2 (the middle number). List the pairs that multiply to -8:
1 × -8 = -8, add: 1 + -8 = -7 ✗
-1 × 8 = -8, add: -1 + 8 = 7 ✗
2 × -4 = -8, add: 2 + -4 = -2 ✗
-2 × 4 = -8, add: -2 + 4 = 2 ✓ this pair works
Put the pair into two brackets: (x − 2)(x + 4).
Check by expanding: x² + 4x − 2x − 8 = x2 + 2x − 8. ✓
Check. Expand your answer: (x − 2)(x + 4) should give x2 + 2x − 8 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?x2 + x − 12
- Factorise. Follow the method.Spot the pattern. This is a quadratic of the form x² + bx + c, with b = 1 and c = -12.
We need two numbers that multiply to -12 (the last number) and add to 1 (the middle number). List the pairs that multiply to -12:
1 × -12 = -12, add: 1 + -12 = -11 ✗
-1 × 12 = -12, add: -1 + 12 = 11 ✗
2 × -6 = -12, add: 2 + -6 = -4 ✗
-2 × 6 = -12, add: -2 + 6 = 4 ✗
3 × -4 = -12, add: 3 + -4 = -1 ✗
-3 × 4 = -12, add: -3 + 4 = 1 ✓ this pair works
Put the pair into two brackets: (x − 3)(x + 4).
Check by expanding: x² + 4x − 3x − 12 = x2 + x − 12. ✓
Check. Expand your answer: (x − 3)(x + 4) should give x2 + x − 12 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?x2 − 3x − 10
- Factorise. Follow the method.Spot the pattern. This is a quadratic of the form x² + bx + c, with b = -3 and c = -10.
We need two numbers that multiply to -10 (the last number) and add to -3 (the middle number). List the pairs that multiply to -10:
1 × -10 = -10, add: 1 + -10 = -9 ✗
-1 × 10 = -10, add: -1 + 10 = 9 ✗
2 × -5 = -10, add: 2 + -5 = -3 ✓ this pair works
Put the pair into two brackets: (x + 2)(x − 5).
Check by expanding: x² − 5x + 2x − 10 = x2 − 3x − 10. ✓
Check. Expand your answer: (x − 5)(x + 2) should give x2 − 3x − 10 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?x2 − 2x − 15
- Factorise. Follow the method.Spot the pattern. This is a quadratic of the form x² + bx + c, with b = -2 and c = -15.
We need two numbers that multiply to -15 (the last number) and add to -2 (the middle number). List the pairs that multiply to -15:
1 × -15 = -15, add: 1 + -15 = -14 ✗
-1 × 15 = -15, add: -1 + 15 = 14 ✗
3 × -5 = -15, add: 3 + -5 = -2 ✓ this pair works
Put the pair into two brackets: (x + 3)(x − 5).
Check by expanding: x² − 5x + 3x − 15 = x2 − 2x − 15. ✓
Check. Expand your answer: (x − 5)(x + 3) should give x2 − 2x − 15 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?x2 + 4x − 21
- Factorise. Follow the method.Spot the pattern. This is a quadratic of the form x² + bx + c, with b = 4 and c = -21.
We need two numbers that multiply to -21 (the last number) and add to 4 (the middle number). List the pairs that multiply to -21:
1 × -21 = -21, add: 1 + -21 = -20 ✗
-1 × 21 = -21, add: -1 + 21 = 20 ✗
3 × -7 = -21, add: 3 + -7 = -4 ✗
-3 × 7 = -21, add: -3 + 7 = 4 ✓ this pair works
Put the pair into two brackets: (x − 3)(x + 7).
Check by expanding: x² + 7x − 3x − 21 = x2 + 4x − 21. ✓
Check. Expand your answer: (x − 3)(x + 7) should give x2 + 4x − 21 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?x2 − x − 20
- Factorise. Follow the method.Spot the pattern. This is a quadratic of the form x² + bx + c, with b = -1 and c = -20.
We need two numbers that multiply to -20 (the last number) and add to -1 (the middle number). List the pairs that multiply to -20:
1 × -20 = -20, add: 1 + -20 = -19 ✗
-1 × 20 = -20, add: -1 + 20 = 19 ✗
2 × -10 = -20, add: 2 + -10 = -8 ✗
-2 × 10 = -20, add: -2 + 10 = 8 ✗
4 × -5 = -20, add: 4 + -5 = -1 ✓ this pair works
Put the pair into two brackets: (x + 4)(x − 5).
Check by expanding: x² − 5x + 4x − 20 = x2 − x − 20. ✓
Check. Expand your answer: (x − 5)(x + 4) should give x2 − x − 20 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?x2 − 9
- Factorise. Follow the method.Spot the pattern. There are two terms, both are perfect squares, and there is a minus between them. This is a difference of two squares.
The rule is: a² − b² = (a − b)(a + b).
Find a and b. x2 = (x)², so a = x. And 9 = 3², so b = 3.
Put them into the rule: (x − 3)(x + 3).
Check by expanding: (x − 3)(x + 3) = x2 + 3x − 3x − 9 = x2 − 9. ✓ (the middle terms cancel)
Check. Expand your answer: (x − 3)(x + 3) should give x2 − 9 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?x2 − 16
- Factorise. Follow the method.Spot the pattern. There are two terms, both are perfect squares, and there is a minus between them. This is a difference of two squares.
The rule is: a² − b² = (a − b)(a + b).
Find a and b. x2 = (x)², so a = x. And 16 = 4², so b = 4.
Put them into the rule: (x − 4)(x + 4).
Check by expanding: (x − 4)(x + 4) = x2 + 4x − 4x − 16 = x2 − 16. ✓ (the middle terms cancel)
Check. Expand your answer: (x − 4)(x + 4) should give x2 − 16 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?x2 − 100
- Factorise. Follow the method.Spot the pattern. There are two terms, both are perfect squares, and there is a minus between them. This is a difference of two squares.
The rule is: a² − b² = (a − b)(a + b).
Find a and b. x2 = (x)², so a = x. And 100 = 10², so b = 10.
Put them into the rule: (x − 10)(x + 10).
Check by expanding: (x − 10)(x + 10) = x2 + 10x − 10x − 100 = x2 − 100. ✓ (the middle terms cancel)
Check. Expand your answer: (x − 10)(x + 10) should give x2 − 100 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?x2 − 1
- Factorise. Follow the method.Spot the pattern. There are two terms, both are perfect squares, and there is a minus between them. This is a difference of two squares.
The rule is: a² − b² = (a − b)(a + b).
Find a and b. x2 = (x)², so a = x. And 1 = 1², so b = 1.
Put them into the rule: (x − 1)(x + 1).
Check by expanding: (x − 1)(x + 1) = x2 + 1x − 1x − 1 = x2 − 1. ✓ (the middle terms cancel)
Check. Expand your answer: (x − 1)(x + 1) should give x2 − 1 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?4x2 − 25
- Factorise. Follow the method.Spot the pattern. There are two terms, both are perfect squares, and there is a minus between them. This is a difference of two squares.
The rule is: a² − b² = (a − b)(a + b).
Find a and b. 4x2 = (2x)², so a = 2x. And 25 = 5², so b = 5.
Put them into the rule: (2x − 5)(2x + 5).
Check by expanding: (2x − 5)(2x + 5) = 4x2 + 52x − 52x − 25 = 4x2 − 25. ✓ (the middle terms cancel)
Check. Expand your answer: (2x − 5)(2x + 5) should give 4x2 − 25 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?9x2 − 4
- Factorise. Follow the method.Spot the pattern. There are two terms, both are perfect squares, and there is a minus between them. This is a difference of two squares.
The rule is: a² − b² = (a − b)(a + b).
Find a and b. 9x2 = (3x)², so a = 3x. And 4 = 2², so b = 2.
Put them into the rule: (3x − 2)(3x + 2).
Check by expanding: (3x − 2)(3x + 2) = 9x2 + 23x − 23x − 4 = 9x2 − 4. ✓ (the middle terms cancel)
Check. Expand your answer: (3x − 2)(3x + 2) should give 9x2 − 4 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Method. Expand the answer she gave and compare it with the question. A correct factorisation expands back to the original.
- Expand Zara's brackets. Multiply each term in the first bracket by each term in the second bracket.First: x × x = x2
Outer: x × 4 = 4x
Inner: 3 × x = 3x
Last: 3 × 4 = 12
(x + 3)(x + 4) = x2 + 7x + 12 - Compare with the question. The question had +8x in the middle, but her brackets give +7x. So she is wrong.7 ≠ 8
- Find the correct numbers. We need two numbers that multiply to 12 and add to 8. Try the factor pairs of 12: 1 and 12 add to 13; 2 and 6 add to 8 (correct); 3 and 4 add to 7 (this was Zara's mistake).2 × 6 = 12 and 2 + 6 = 8
- Write the correct answer. Put the two numbers into the brackets.(x + 2)(x + 6)
Check. Expand: x2 + 6x + 2x + 12 = x2 + 8x + 12. ✓
Watch out. Zara found two numbers that multiply to 12 but did not check that they add to 8. Always check both conditions.
x2 + 6x + 9, x2 − 25, x2 + 4x + 5, x2 − 5x + 6
Method. Expand each one, or test whether a pair of numbers exists that multiplies to the last number and adds to the middle number.
- x2 + 6x + 9. Numbers that multiply to 9 and add to 6: 3 and 3.= (x + 3)2. It can be factorised.
- x2 − 25. A minus between two squares: difference of two squares.= (x − 5)(x + 5). It can be factorised.
- x2 + 4x + 5. Pairs that multiply to 5: only 1 and 5, and 1 + 5 = 6, not 4. Negative pairs add to −6.No pair works, so it cannot be factorised.
- x2 − 5x + 6. Numbers that multiply to 6 and add to −5: −2 and −3.= (x − 2)(x − 3). It can be factorised.
Check. Expanding (x + 3)2, (x − 5)(x + 5) and (x − 2)(x − 3) gives the other three expressions. ✓
Method. Factorise the area. The area of a rectangle is length × width, so the two brackets are the length and the width.
- Factorise the area. We need two numbers that multiply to 20 and add to 9.Pairs for 20: 1 and 20 (21), 2 and 10 (12), 4 and 5 (9)
x2 + 9x + 20 = (x + 4)(x + 5) - Match with the width. One bracket is the width, (x + 4). The other bracket must be the length.Length = x + 5
Check. (x + 4)(x + 5) = x2 + 5x + 4x + 20 = x2 + 9x + 20. ✓
Watch out. Write the units: the length is (x + 5) cm.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?x2 + 12x + 35
- Factorise. Follow the method.Spot the pattern. This is a quadratic of the form x² + bx + c, with b = 12 and c = 35.
We need two numbers that multiply to 35 (the last number) and add to 12 (the middle number). List the pairs that multiply to 35:
1 × 35 = 35, add: 1 + 35 = 36 ✗
-1 × -35 = 35, add: -1 + -35 = -36 ✗
5 × 7 = 35, add: 5 + 7 = 12 ✓ this pair works
Put the pair into two brackets: (x + 5)(x + 7).
Check by expanding: x² + 7x + 5x + 35 = x2 + 12x + 35. ✓
Check. Expand your answer: (x + 5)(x + 7) should give x2 + 12x + 35 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?x2 − 4x
- Factorise. Follow the method.Look for a common factor. A common factor is something that divides into every term.
The terms are x2, −4x. The biggest number and the highest power of x that divide into all of them is x.
Take it outside a bracket, and inside the bracket put each term divided by it:
x2 ÷ x = x
−4x ÷ x = −4
So x2 − 4x = x(x − 4).
Check by expanding: x × (x − 4) = x × x, x × (−4) → x2 − 4x. ✓
Check. Expand your answer: (x − 4)x should give x2 − 4x again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?7x − 21
- Factorise. Follow the method.Look for a common factor. A common factor is something that divides into every term.
The terms are 7x, −21. The biggest number and the highest power of x that divide into all of them is 7.
Take it outside a bracket, and inside the bracket put each term divided by it:
7x ÷ 7 = x
−21 ÷ 7 = −3
So 7x − 21 = 7(x − 3).
Check by expanding: 7 × (x − 3) = 7 × x, 7 × (−3) → 7x − 21. ✓
Check. Expand your answer: 7(x − 3) should give 7x − 21 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?x2 − 144
- Factorise. Follow the method.Spot the pattern. There are two terms, both are perfect squares, and there is a minus between them. This is a difference of two squares.
The rule is: a² − b² = (a − b)(a + b).
Find a and b. x2 = (x)², so a = x. And 144 = 12², so b = 12.
Put them into the rule: (x − 12)(x + 12).
Check by expanding: (x − 12)(x + 12) = x2 + 12x − 12x − 144 = x2 − 144. ✓ (the middle terms cancel)
Check. Expand your answer: (x − 12)(x + 12) should give x2 − 144 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?x2 − 11x + 28
- Factorise. Follow the method.Spot the pattern. This is a quadratic of the form x² + bx + c, with b = -11 and c = 28.
We need two numbers that multiply to 28 (the last number) and add to -11 (the middle number). List the pairs that multiply to 28:
1 × 28 = 28, add: 1 + 28 = 29 ✗
-1 × -28 = 28, add: -1 + -28 = -29 ✗
2 × 14 = 28, add: 2 + 14 = 16 ✗
-2 × -14 = 28, add: -2 + -14 = -16 ✗
4 × 7 = 28, add: 4 + 7 = 11 ✗
-4 × -7 = 28, add: -4 + -7 = -11 ✓ this pair works
Put the pair into two brackets: (x − 4)(x − 7).
Check by expanding: x² − 7x − 4x + 28 = x2 − 11x + 28. ✓
Check. Expand your answer: (x − 7)(x − 4) should give x2 − 11x + 28 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?x2 + 3x − 28
- Factorise. Follow the method.Spot the pattern. This is a quadratic of the form x² + bx + c, with b = 3 and c = -28.
We need two numbers that multiply to -28 (the last number) and add to 3 (the middle number). List the pairs that multiply to -28:
1 × -28 = -28, add: 1 + -28 = -27 ✗
-1 × 28 = -28, add: -1 + 28 = 27 ✗
2 × -14 = -28, add: 2 + -14 = -12 ✗
-2 × 14 = -28, add: -2 + 14 = 12 ✗
4 × -7 = -28, add: 4 + -7 = -3 ✗
-4 × 7 = -28, add: -4 + 7 = 3 ✓ this pair works
Put the pair into two brackets: (x − 4)(x + 7).
Check by expanding: x² + 7x − 4x − 28 = x2 + 3x − 28. ✓
Check. Expand your answer: (x − 4)(x + 7) should give x2 + 3x − 28 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Method. Use the difference of two squares, a2 − b2 = (a − b)(a + b), with a and b as the two numbers.
- Write the question as a difference of two squares. There is a minus between two squared numbers.522 − 482
- Use the rule. Replace a by 52 and b by 48.(52 − 48)(52 + 48)
- Work out each bracket, then multiply. The brackets are now easy to do in your head.4 × 100 = 400
Check. 522 = 2704 and 482 = 2304, and 2704 − 2304 = 400. ✓
Watch out. This only works with a minus between the squares.
Method. Use the difference of two squares again.
- Write it with a = 101 and b = 99. The rule is a2 − b2 = (a − b)(a + b).(101 − 99)(101 + 99)
- Work out each bracket. 101 − 99 = 2 and 101 + 99 = 200.2 × 200 = 400
Check. 1012 = 10201, 992 = 9801 and 10201 − 9801 = 400. ✓
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?2x2 + 7x + 3
- Factorise. Follow the method.Spot the pattern. A quadratic where the number in front of x² is 2, not 1. We use the split the middle term method.
Multiply the first and last numbers: 2 × (3) = 6. We need two numbers that multiply to 6 and add to 7 (the middle number).
1 × 6 = 6, add: 7 ✓ this pair works
Split the middle term (7x) into 1x and 6x: 2x2 + 1x + 6x + 3.
Factorise each pair separately. First pair: 2x2 + x = x(2x + 1). Second pair: 6x + 3 = 3(2x + 1).
The two brackets (2x + 1) are the same. That is how we know it has worked. Write: x(2x + 1) + 3(2x + 1).
Take the common bracket out: (2x + 1) multiplied by (x + 3).
Result: (x + 3)(2x + 1).
Check by expanding: (x + 3)(2x + 1) = 2x2 + 7x + 3. ✓
Check. Expand your answer: (x + 3)(2x + 1) should give 2x2 + 7x + 3 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?3x2 + 10x + 8
- Factorise. Follow the method.Spot the pattern. A quadratic where the number in front of x² is 3, not 1. We use the split the middle term method.
Multiply the first and last numbers: 3 × (8) = 24. We need two numbers that multiply to 24 and add to 10 (the middle number).
1 × 24 = 24, add: 25 ✗
-1 × -24 = 24, add: -25 ✗
2 × 12 = 24, add: 14 ✗
-2 × -12 = 24, add: -14 ✗
3 × 8 = 24, add: 11 ✗
-3 × -8 = 24, add: -11 ✗
4 × 6 = 24, add: 10 ✓ this pair works
Split the middle term (10x) into 4x and 6x: 3x2 + 4x + 6x + 8.
Factorise each pair separately. First pair: 3x2 + 4x = x(3x + 4). Second pair: 6x + 8 = 2(3x + 4).
The two brackets (3x + 4) are the same. That is how we know it has worked. Write: x(3x + 4) + 2(3x + 4).
Take the common bracket out: (3x + 4) multiplied by (x + 2).
Result: (x + 2)(3x + 4).
Check by expanding: (x + 2)(3x + 4) = 3x2 + 10x + 8. ✓
Check. Expand your answer: (x + 2)(3x + 4) should give 3x2 + 10x + 8 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?2x2 − 5x − 3
- Factorise. Follow the method.Spot the pattern. A quadratic where the number in front of x² is 2, not 1. We use the split the middle term method.
Multiply the first and last numbers: 2 × (-3) = -6. We need two numbers that multiply to -6 and add to -5 (the middle number).
1 × -6 = -6, add: -5 ✓ this pair works
Split the middle term (-5x) into 1x and -6x: 2x2 + 1x − 6x − 3.
Factorise each pair separately. First pair: 2x2 + x = x(2x + 1). Second pair: −6x − 3 = −3(2x + 1).
The two brackets (2x + 1) are the same. That is how we know it has worked. Write: x(2x + 1) − 3(2x + 1).
Take the common bracket out: (2x + 1) multiplied by (x − 3).
Result: (x − 3)(2x + 1).
Check by expanding: (x − 3)(2x + 1) = 2x2 − 5x − 3. ✓
Check. Expand your answer: (x − 3)(2x + 1) should give 2x2 − 5x − 3 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?5x2 − 11x + 2
- Factorise. Follow the method.Spot the pattern. A quadratic where the number in front of x² is 5, not 1. We use the split the middle term method.
Multiply the first and last numbers: 5 × (2) = 10. We need two numbers that multiply to 10 and add to -11 (the middle number).
1 × 10 = 10, add: 11 ✗
-1 × -10 = 10, add: -11 ✓ this pair works
Split the middle term (-11x) into -1x and -10x: 5x2 − 1x − 10x + 2.
Factorise each pair separately. First pair: 5x2 − x = x(5x − 1). Second pair: −10x + 2 = 2(−5x + 1).
The two brackets (5x − 1) are the same. That is how we know it has worked. Write: x(5x − 1) + 2(−5x + 1).
Take the common bracket out: (5x − 1) multiplied by (x + 2).
Result: (x − 2)(5x − 1).
Check by expanding: (x − 2)(5x − 1) = 5x2 − 11x + 2. ✓
Check. Expand your answer: (x − 2)(5x − 1) should give 5x2 − 11x + 2 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?6x2 + x − 2
- Factorise. Follow the method.Spot the pattern. A quadratic where the number in front of x² is 6, not 1. We use the split the middle term method.
Multiply the first and last numbers: 6 × (-2) = -12. We need two numbers that multiply to -12 and add to 1 (the middle number).
1 × -12 = -12, add: -11 ✗
-1 × 12 = -12, add: 11 ✗
2 × -6 = -12, add: -4 ✗
-2 × 6 = -12, add: 4 ✗
3 × -4 = -12, add: -1 ✗
-3 × 4 = -12, add: 1 ✓ this pair works
Split the middle term (1x) into -3x and 4x: 6x2 − 3x + 4x − 2.
Factorise each pair separately. First pair: 6x2 − 3x = 3x(2x − 1). Second pair: 4x − 2 = 2(2x − 1).
The two brackets (2x − 1) are the same. That is how we know it has worked. Write: 3x(2x − 1) + 2(2x − 1).
Take the common bracket out: (2x − 1) multiplied by (3x + 2).
Result: (2x − 1)(3x + 2).
Check by expanding: (2x − 1)(3x + 2) = 6x2 + x − 2. ✓
Check. Expand your answer: (2x − 1)(3x + 2) should give 6x2 + x − 2 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?4x2 − 4x − 3
- Factorise. Follow the method.Spot the pattern. A quadratic where the number in front of x² is 4, not 1. We use the split the middle term method.
Multiply the first and last numbers: 4 × (-3) = -12. We need two numbers that multiply to -12 and add to -4 (the middle number).
1 × -12 = -12, add: -11 ✗
-1 × 12 = -12, add: 11 ✗
2 × -6 = -12, add: -4 ✓ this pair works
Split the middle term (-4x) into 2x and -6x: 4x2 + 2x − 6x − 3.
Factorise each pair separately. First pair: 4x2 + 2x = 2x(2x + 1). Second pair: −6x − 3 = −3(2x + 1).
The two brackets (2x + 1) are the same. That is how we know it has worked. Write: 2x(2x + 1) − 3(2x + 1).
Take the common bracket out: (2x + 1) multiplied by (2x − 3).
Result: (2x − 3)(2x + 1).
Check by expanding: (2x − 3)(2x + 1) = 4x2 − 4x − 3. ✓
Check. Expand your answer: (2x − 3)(2x + 1) should give 4x2 − 4x − 3 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?2x2 − 50
- Factorise. Follow the method.Look for a common factor. A common factor is something that divides into every term.
The terms are 2x2, −50. The biggest number and the highest power of x that divide into all of them is 2.
Take it outside a bracket, and inside the bracket put each term divided by it:
2x2 ÷ 2 = x2
−50 ÷ 2 = −25
So 2x2 − 50 = 2(x2 − 25).
Check by expanding: 2 × (x2 − 25) = 2 × x2, 2 × (−25) → 2x2 − 50. ✓
Can the bracket be factorised again? Look at (x2 − 25).
Spot the pattern. There are two terms, both are perfect squares, and there is a minus between them. This is a difference of two squares.
The rule is: a² − b² = (a − b)(a + b).
Find a and b. x2 = (x)², so a = x. And 25 = 5², so b = 5.
Put them into the rule: (x − 5)(x + 5).
Check by expanding: (x − 5)(x + 5) = x2 + 5x − 5x − 25 = x2 − 25. ✓ (the middle terms cancel)
Check. Expand your answer: 2(x − 5)(x + 5) should give 2x2 − 50 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?3x2 + 15x + 18
- Factorise. Follow the method.Look for a common factor. A common factor is something that divides into every term.
The terms are 3x2, 15x, 18. The biggest number and the highest power of x that divide into all of them is 3.
Take it outside a bracket, and inside the bracket put each term divided by it:
3x2 ÷ 3 = x2
15x ÷ 3 = 5x
18 ÷ 3 = 6
So 3x2 + 15x + 18 = 3(x2 + 5x + 6).
Check by expanding: 3 × (x2 + 5x + 6) = 3 × x2, 3 × 5x, 3 × 6 → 3x2 + 15x + 18. ✓
Can the bracket be factorised again? Look at (x2 + 5x + 6).
Spot the pattern. This is a quadratic of the form x² + bx + c, with b = 5 and c = 6.
We need two numbers that multiply to 6 (the last number) and add to 5 (the middle number). List the pairs that multiply to 6:
1 × 6 = 6, add: 1 + 6 = 7 ✗
-1 × -6 = 6, add: -1 + -6 = -7 ✗
2 × 3 = 6, add: 2 + 3 = 5 ✓ this pair works
Put the pair into two brackets: (x + 2)(x + 3).
Check by expanding: x² + 3x + 2x + 6 = x2 + 5x + 6. ✓
Check. Expand your answer: 3(x + 2)(x + 3) should give 3x2 + 15x + 18 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?5x2 − 45
- Factorise. Follow the method.Look for a common factor. A common factor is something that divides into every term.
The terms are 5x2, −45. The biggest number and the highest power of x that divide into all of them is 5.
Take it outside a bracket, and inside the bracket put each term divided by it:
5x2 ÷ 5 = x2
−45 ÷ 5 = −9
So 5x2 − 45 = 5(x2 − 9).
Check by expanding: 5 × (x2 − 9) = 5 × x2, 5 × (−9) → 5x2 − 45. ✓
Can the bracket be factorised again? Look at (x2 − 9).
Spot the pattern. There are two terms, both are perfect squares, and there is a minus between them. This is a difference of two squares.
The rule is: a² − b² = (a − b)(a + b).
Find a and b. x2 = (x)², so a = x. And 9 = 3², so b = 3.
Put them into the rule: (x − 3)(x + 3).
Check by expanding: (x − 3)(x + 3) = x2 + 3x − 3x − 9 = x2 − 9. ✓ (the middle terms cancel)
Check. Expand your answer: 5(x − 3)(x + 3) should give 5x2 − 45 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?2x2 − 12x + 16
- Factorise. Follow the method.Look for a common factor. A common factor is something that divides into every term.
The terms are 2x2, −12x, 16. The biggest number and the highest power of x that divide into all of them is 2.
Take it outside a bracket, and inside the bracket put each term divided by it:
2x2 ÷ 2 = x2
−12x ÷ 2 = −6x
16 ÷ 2 = 8
So 2x2 − 12x + 16 = 2(x2 − 6x + 8).
Check by expanding: 2 × (x2 − 6x + 8) = 2 × x2, 2 × (−6x), 2 × 8 → 2x2 − 12x + 16. ✓
Can the bracket be factorised again? Look at (x2 − 6x + 8).
Spot the pattern. This is a quadratic of the form x² + bx + c, with b = -6 and c = 8.
We need two numbers that multiply to 8 (the last number) and add to -6 (the middle number). List the pairs that multiply to 8:
1 × 8 = 8, add: 1 + 8 = 9 ✗
-1 × -8 = 8, add: -1 + -8 = -9 ✗
2 × 4 = 8, add: 2 + 4 = 6 ✗
-2 × -4 = 8, add: -2 + -4 = -6 ✓ this pair works
Put the pair into two brackets: (x − 2)(x − 4).
Check by expanding: x² − 4x − 2x + 8 = x2 − 6x + 8. ✓
Check. Expand your answer: 2(x − 4)(x − 2) should give 2x2 − 12x + 16 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?6x2 + 9x
- Factorise. Follow the method.Look for a common factor. A common factor is something that divides into every term.
The terms are 6x2, 9x. The biggest number and the highest power of x that divide into all of them is 3x.
Take it outside a bracket, and inside the bracket put each term divided by it:
6x2 ÷ 3x = 2x
9x ÷ 3x = 3
So 6x2 + 9x = 3x(2x + 3).
Check by expanding: 3x × (2x + 3) = 3x × 2x, 3x × 3 → 6x2 + 9x. ✓
Check. Expand your answer: 3x(2x + 3) should give 6x2 + 9x again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Key words. To factorise is the opposite of expanding: we turn a sum like x² + 5x + 6 into a product like (x + 2)(x + 3).
Method. Factorising means writing an expression as a product (brackets multiplied together).
- Look at the expression and decide which method to use. Check in this order: (1) is there a common factor? (2) is it a difference of two squares? (3) is it a quadratic with three terms?4x2 − 36
- Factorise. Follow the method.Look for a common factor. A common factor is something that divides into every term.
The terms are 4x2, −36. The biggest number and the highest power of x that divide into all of them is 4.
Take it outside a bracket, and inside the bracket put each term divided by it:
4x2 ÷ 4 = x2
−36 ÷ 4 = −9
So 4x2 − 36 = 4(x2 − 9).
Check by expanding: 4 × (x2 − 9) = 4 × x2, 4 × (−9) → 4x2 − 36. ✓
Can the bracket be factorised again? Look at (x2 − 9).
Spot the pattern. There are two terms, both are perfect squares, and there is a minus between them. This is a difference of two squares.
The rule is: a² − b² = (a − b)(a + b).
Find a and b. x2 = (x)², so a = x. And 9 = 3², so b = 3.
Put them into the rule: (x − 3)(x + 3).
Check by expanding: (x − 3)(x + 3) = x2 + 3x − 3x − 9 = x2 − 9. ✓ (the middle terms cancel)
Check. Expand your answer: 4(x − 3)(x + 3) should give 4x2 − 36 again. ✓
Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong.
Method. Treat (2n + 1) as a and (2n − 1) as b and use the difference of two squares.
- Write the rule. a2 − b2 = (a − b)(a + b)Here a = 2n + 1 and b = 2n − 1.
- Work out a − b. Take care: subtracting a bracket changes both signs.(2n + 1) − (2n − 1) = 2n + 1 − 2n + 1 = 2
- Work out a + b. Adding: the 2n terms add and the numbers cancel.(2n + 1) + (2n − 1) = 4n
- Multiply. The two brackets are multiplied together.2 × 4n = 8n
- Conclude. 8n is 8 multiplied by a whole number, so it is a multiple of 8.The difference is a multiple of 8 for every whole number n.
Check. Try n = 3: 72 − 52 = 49 − 25 = 24 = 8 × 3. ✓
Watch out. In a proof, say clearly why the result is a multiple of 8: it has a factor of 8.
Method. x2 + bx + 12 factorises when there are two whole numbers that multiply to 12. The number b is then their sum. List every pair.
- List the factor pairs of 12, including negative pairs. The numbers must multiply to +12, so they have the same sign: both positive or both negative.1 × 12, 2 × 6, 3 × 4
(−1) × (−12), (−2) × (−6), (−3) × (−4) - Add each pair to find b. The sum of the pair is b.1 + 12 = 13, 2 + 6 = 8, 3 + 4 = 7
and −13, −8, −7 for the negative pairs - Count the different values. All six sums are different.b = −13, −8, −7, 7, 8, 13: 6 values
Check. For example b = −8: x2 − 8x + 12 = (x − 2)(x − 6). ✓
Watch out. Do not forget the negative pairs: both numbers can be negative, which gives a negative b.