GCSE Maths: Factorising ax² + bx + c (Higher)

Key words. A factor of a term divides into it exactly. A common factor of several terms is a factor of all of them. To factorise an expression means to write it as a product, with brackets. It is the reverse of expanding.

Method. Factorising: (1) take out any common factor, (2) look at what is left: a difference of two squares or a quadratic with three terms, (3) check by expanding.

1. Check for a common factor first. Always check this first. A common factor is a number or letter that divides into every term.
   The terms of 2x² + 5x + 3 have no common factor other than 1, so there is nothing to take out.
2. Factorise the quadratic. It has three terms, so we use the "two numbers" method. Follow every line carefully.
   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: 7 ✗
   −1 × −6 = 6,   add: −7 ✗
   2 × 3 = 6,   add: 5 ✓ this pair works
   Split the middle term (5x) into 2x and 3x: 2x² + 2x + 3x + 3.
   Factorise each pair separately. First pair: 2x² + 2x = 2x(x + 1). Second pair: 3x + 3 = 3(x + 1).
   The two brackets (x + 1) are the same. That is how we know it has worked. Write: 2x(x + 1) + 3(x + 1).
   Take the common bracket out: (x + 1) multiplied by (2x + 3).
   Result: (x + 1)(2x + 3).
   Check by expanding: (x + 1)(2x + 3) = 2x² + 5x + 3. ✓
3. Write the answer. The brackets are the factors of the quadratic. We have checked by expanding that they multiply back to the original.
   2x² + 5x + 3 = (x + 1)(2x + 3)

Check. Expand the answer: (x + 1)(2x + 3) gives 2x² + 5x + 3 again. ✓

Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong. And check for a common factor first: it makes the numbers smaller.

Key words. A factor of a term divides into it exactly. A common factor of several terms is a factor of all of them. To factorise an expression means to write it as a product, with brackets. It is the reverse of expanding.

Method. Factorising: (1) take out any common factor, (2) look at what is left: a difference of two squares or a quadratic with three terms, (3) check by expanding.

1. Check for a common factor first. Always check this first. A common factor is a number or letter that divides into every term.
   The terms of 2x² + 7x + 6 have no common factor other than 1, so there is nothing to take out.
2. Factorise the quadratic. It has three terms, so we use the "two numbers" method. Follow every line carefully.
   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 × (6) = 12. We need two numbers that multiply to 12 and add to 7 (the middle number).
   1 × 12 = 12,   add: 13 ✗
   −1 × −12 = 12,   add: −13 ✗
   2 × 6 = 12,   add: 8 ✗
   −2 × −6 = 12,   add: −8 ✗
   3 × 4 = 12,   add: 7 ✓ this pair works
   Split the middle term (7x) into 3x and 4x: 2x² + 3x + 4x + 6.
   Factorise each pair separately. First pair: 2x² + 3x = x(2x + 3). Second pair: 4x + 6 = 2(2x + 3).
   The two brackets (2x + 3) are the same. That is how we know it has worked. Write: x(2x + 3) + 2(2x + 3).
   Take the common bracket out: (2x + 3) multiplied by (x + 2).
   Result: (x + 2)(2x + 3).
   Check by expanding: (x + 2)(2x + 3) = 2x² + 7x + 6. ✓
3. Write the answer. The brackets are the factors of the quadratic. We have checked by expanding that they multiply back to the original.
   2x² + 7x + 6 = (x + 2)(2x + 3)

Check. Expand the answer: (x + 2)(2x + 3) gives 2x² + 7x + 6 again. ✓

Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong. And check for a common factor first: it makes the numbers smaller.

Key words. A factor of a term divides into it exactly. A common factor of several terms is a factor of all of them. To factorise an expression means to write it as a product, with brackets. It is the reverse of expanding.

Method. Factorising: (1) take out any common factor, (2) look at what is left: a difference of two squares or a quadratic with three terms, (3) check by expanding.

1. Check for a common factor first. Always check this first. A common factor is a number or letter that divides into every term.
   The terms of 3x² + 7x + 2 have no common factor other than 1, so there is nothing to take out.
2. Factorise the quadratic. It has three terms, so we use the "two numbers" method. Follow every line carefully.
   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 × (2) = 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: 3x² + 1x + 6x + 2.
   Factorise each pair separately. First pair: 3x² + x = x(3x + 1). Second pair: 6x + 2 = 2(3x + 1).
   The two brackets (3x + 1) are the same. That is how we know it has worked. Write: x(3x + 1) + 2(3x + 1).
   Take the common bracket out: (3x + 1) multiplied by (x + 2).
   Result: (x + 2)(3x + 1).
   Check by expanding: (x + 2)(3x + 1) = 3x² + 7x + 2. ✓
3. Write the answer. The brackets are the factors of the quadratic. We have checked by expanding that they multiply back to the original.
   3x² + 7x + 2 = (x + 2)(3x + 1)

Check. Expand the answer: (x + 2)(3x + 1) gives 3x² + 7x + 2 again. ✓

Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong. And check for a common factor first: it makes the numbers smaller.

Key words. A factor of a term divides into it exactly. A common factor of several terms is a factor of all of them. To factorise an expression means to write it as a product, with brackets. It is the reverse of expanding.

Method. Factorising: (1) take out any common factor, (2) look at what is left: a difference of two squares or a quadratic with three terms, (3) check by expanding.

1. Check for a common factor first. Always check this first. A common factor is a number or letter that divides into every term.
   The terms of 5x² + 16x + 3 have no common factor other than 1, so there is nothing to take out.
2. Factorise the quadratic. It has three terms, so we use the "two numbers" method. Follow every line carefully.
   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 × (3) = 15. We need two numbers that multiply to 15 and add to 16 (the middle number).
   1 × 15 = 15,   add: 16 ✓ this pair works
   Split the middle term (16x) into 1x and 15x: 5x² + 1x + 15x + 3.
   Factorise each pair separately. First pair: 5x² + x = x(5x + 1). Second pair: 15x + 3 = 3(5x + 1).
   The two brackets (5x + 1) are the same. That is how we know it has worked. Write: x(5x + 1) + 3(5x + 1).
   Take the common bracket out: (5x + 1) multiplied by (x + 3).
   Result: (x + 3)(5x + 1).
   Check by expanding: (x + 3)(5x + 1) = 5x² + 16x + 3. ✓
3. Write the answer. The brackets are the factors of the quadratic. We have checked by expanding that they multiply back to the original.
   5x² + 16x + 3 = (x + 3)(5x + 1)

Check. Expand the answer: (x + 3)(5x + 1) gives 5x² + 16x + 3 again. ✓

Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong. And check for a common factor first: it makes the numbers smaller.

Key words. A factor of a term divides into it exactly. A common factor of several terms is a factor of all of them. To factorise an expression means to write it as a product, with brackets. It is the reverse of expanding.

Method. Factorising: (1) take out any common factor, (2) look at what is left: a difference of two squares or a quadratic with three terms, (3) check by expanding.

1. Check for a common factor first. Always check this first. A common factor is a number or letter that divides into every term.
   The terms of 2x² + 9x + 10 have no common factor other than 1, so there is nothing to take out.
2. Factorise the quadratic. It has three terms, so we use the "two numbers" method. Follow every line carefully.
   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 × (10) = 20. We need two numbers that multiply to 20 and add to 9 (the middle number).
   1 × 20 = 20,   add: 21 ✗
   −1 × −20 = 20,   add: −21 ✗
   2 × 10 = 20,   add: 12 ✗
   −2 × −10 = 20,   add: −12 ✗
   4 × 5 = 20,   add: 9 ✓ this pair works
   Split the middle term (9x) into 4x and 5x: 2x² + 4x + 5x + 10.
   Factorise each pair separately. First pair: 2x² + 4x = 2x(x + 2). Second pair: 5x + 10 = 5(x + 2).
   The two brackets (x + 2) are the same. That is how we know it has worked. Write: 2x(x + 2) + 5(x + 2).
   Take the common bracket out: (x + 2) multiplied by (2x + 5).
   Result: (x + 2)(2x + 5).
   Check by expanding: (x + 2)(2x + 5) = 2x² + 9x + 10. ✓
3. Write the answer. The brackets are the factors of the quadratic. We have checked by expanding that they multiply back to the original.
   2x² + 9x + 10 = (x + 2)(2x + 5)

Check. Expand the answer: (x + 2)(2x + 5) gives 2x² + 9x + 10 again. ✓

Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong. And check for a common factor first: it makes the numbers smaller.

Key words. A factor of a term divides into it exactly. A common factor of several terms is a factor of all of them. To factorise an expression means to write it as a product, with brackets. It is the reverse of expanding.

Method. Factorising: (1) take out any common factor, (2) look at what is left: a difference of two squares or a quadratic with three terms, (3) check by expanding.

1. Check for a common factor first. Always check this first. A common factor is a number or letter that divides into every term.
   The terms of 3x² + 11x + 6 have no common factor other than 1, so there is nothing to take out.
2. Factorise the quadratic. It has three terms, so we use the "two numbers" method. Follow every line carefully.
   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 × (6) = 18. We need two numbers that multiply to 18 and add to 11 (the middle number).
   1 × 18 = 18,   add: 19 ✗
   −1 × −18 = 18,   add: −19 ✗
   2 × 9 = 18,   add: 11 ✓ this pair works
   Split the middle term (11x) into 2x and 9x: 3x² + 2x + 9x + 6.
   Factorise each pair separately. First pair: 3x² + 2x = x(3x + 2). Second pair: 9x + 6 = 3(3x + 2).
   The two brackets (3x + 2) are the same. That is how we know it has worked. Write: x(3x + 2) + 3(3x + 2).
   Take the common bracket out: (3x + 2) multiplied by (x + 3).
   Result: (x + 3)(3x + 2).
   Check by expanding: (x + 3)(3x + 2) = 3x² + 11x + 6. ✓
3. Write the answer. The brackets are the factors of the quadratic. We have checked by expanding that they multiply back to the original.
   3x² + 11x + 6 = (x + 3)(3x + 2)

Check. Expand the answer: (x + 3)(3x + 2) gives 3x² + 11x + 6 again. ✓

Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong. And check for a common factor first: it makes the numbers smaller.

Key words. A factor of a term divides into it exactly. A common factor of several terms is a factor of all of them. To factorise an expression means to write it as a product, with brackets. It is the reverse of expanding.

Method. Factorising: (1) take out any common factor, (2) look at what is left: a difference of two squares or a quadratic with three terms, (3) check by expanding.

1. Check for a common factor first. Always check this first. A common factor is a number or letter that divides into every term.
   The terms of 2x² − 7x + 3 have no common factor other than 1, so there is nothing to take out.
2. Factorise the quadratic. It has three terms, so we use the "two numbers" method. Follow every line carefully.
   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 ✗
   −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. ✓
3. Write the answer. The brackets are the factors of the quadratic. We have checked by expanding that they multiply back to the original.
   2x² − 7x + 3 = (x − 3)(2x − 1)

Check. Expand the answer: (x − 3)(2x − 1) gives 2x² − 7x + 3 again. ✓

Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong. And check for a common factor first: it makes the numbers smaller.

Key words. A factor of a term divides into it exactly. A common factor of several terms is a factor of all of them. To factorise an expression means to write it as a product, with brackets. It is the reverse of expanding.

Method. Factorising: (1) take out any common factor, (2) look at what is left: a difference of two squares or a quadratic with three terms, (3) check by expanding.

1. Check for a common factor first. Always check this first. A common factor is a number or letter that divides into every term.
   The terms of 3x² − 8x + 4 have no common factor other than 1, so there is nothing to take out.
2. Factorise the quadratic. It has three terms, so we use the "two numbers" method. Follow every line carefully.
   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 × (4) = 12. We need two numbers that multiply to 12 and add to −8 (the middle number).
   1 × 12 = 12,   add: 13 ✗
   −1 × −12 = 12,   add: −13 ✗
   2 × 6 = 12,   add: 8 ✗
   −2 × −6 = 12,   add: −8 ✓ this pair works
   Split the middle term (−8x) into −2x and −6x: 3x² − 2x − 6x + 4.
   Factorise each pair separately. First pair: 3x² − 2x = x(3x − 2). Second pair: −6x + 4 = 2(−3x + 2).
   The two brackets (3x − 2) are the same. That is how we know it has worked. Write: x(3x − 2) + 2(−3x + 2).
   Take the common bracket out: (3x − 2) multiplied by (x + 2).
   Result: (x − 2)(3x − 2).
   Check by expanding: (x − 2)(3x − 2) = 3x² − 8x + 4. ✓
3. Write the answer. The brackets are the factors of the quadratic. We have checked by expanding that they multiply back to the original.
   3x² − 8x + 4 = (x − 2)(3x − 2)

Check. Expand the answer: (x − 2)(3x − 2) gives 3x² − 8x + 4 again. ✓

Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong. And check for a common factor first: it makes the numbers smaller.

Key words. A factor of a term divides into it exactly. A common factor of several terms is a factor of all of them. To factorise an expression means to write it as a product, with brackets. It is the reverse of expanding.

Method. Factorising: (1) take out any common factor, (2) look at what is left: a difference of two squares or a quadratic with three terms, (3) check by expanding.

1. Check for a common factor first. Always check this first. A common factor is a number or letter that divides into every term.
   The terms of 5x² − 13x + 6 have no common factor other than 1, so there is nothing to take out.
2. Factorise the quadratic. It has three terms, so we use the "two numbers" method. Follow every line carefully.
   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 × (6) = 30. We need two numbers that multiply to 30 and add to −13 (the middle number).
   1 × 30 = 30,   add: 31 ✗
   −1 × −30 = 30,   add: −31 ✗
   2 × 15 = 30,   add: 17 ✗
   −2 × −15 = 30,   add: −17 ✗
   3 × 10 = 30,   add: 13 ✗
   −3 × −10 = 30,   add: −13 ✓ this pair works
   Split the middle term (−13x) into −3x and −10x: 5x² − 3x − 10x + 6.
   Factorise each pair separately. First pair: 5x² − 3x = x(5x − 3). Second pair: −10x + 6 = 2(−5x + 3).
   The two brackets (5x − 3) are the same. That is how we know it has worked. Write: x(5x − 3) + 2(−5x + 3).
   Take the common bracket out: (5x − 3) multiplied by (x + 2).
   Result: (x − 2)(5x − 3).
   Check by expanding: (x − 2)(5x − 3) = 5x² − 13x + 6. ✓
3. Write the answer. The brackets are the factors of the quadratic. We have checked by expanding that they multiply back to the original.
   5x² − 13x + 6 = (x − 2)(5x − 3)

Check. Expand the answer: (x − 2)(5x − 3) gives 5x² − 13x + 6 again. ✓

Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong. And check for a common factor first: it makes the numbers smaller.

Key words. A factor of a term divides into it exactly. A common factor of several terms is a factor of all of them. To factorise an expression means to write it as a product, with brackets. It is the reverse of expanding.

Method. Factorising: (1) take out any common factor, (2) look at what is left: a difference of two squares or a quadratic with three terms, (3) check by expanding.

1. Check for a common factor first. Always check this first. A common factor is a number or letter that divides into every term.
   The terms of 2x² − 11x + 12 have no common factor other than 1, so there is nothing to take out.
2. Factorise the quadratic. It has three terms, so we use the "two numbers" method. Follow every line carefully.
   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 × (12) = 24. We need two numbers that multiply to 24 and add to −11 (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 ✓ this pair works
   Split the middle term (−11x) into −3x and −8x: 2x² − 3x − 8x + 12.
   Factorise each pair separately. First pair: 2x² − 3x = x(2x − 3). Second pair: −8x + 12 = 4(−2x + 3).
   The two brackets (2x − 3) are the same. That is how we know it has worked. Write: x(2x − 3) + 4(−2x + 3).
   Take the common bracket out: (2x − 3) multiplied by (x + 4).
   Result: (x − 4)(2x − 3).
   Check by expanding: (x − 4)(2x − 3) = 2x² − 11x + 12. ✓
3. Write the answer. The brackets are the factors of the quadratic. We have checked by expanding that they multiply back to the original.
   2x² − 11x + 12 = (x − 4)(2x − 3)

Check. Expand the answer: (x − 4)(2x − 3) gives 2x² − 11x + 12 again. ✓

Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong. And check for a common factor first: it makes the numbers smaller.

Key words. A factor of a term divides into it exactly. A common factor of several terms is a factor of all of them. To factorise an expression means to write it as a product, with brackets. It is the reverse of expanding.

Method. Factorising: (1) take out any common factor, (2) look at what is left: a difference of two squares or a quadratic with three terms, (3) check by expanding.

1. Check for a common factor first. Always check this first. A common factor is a number or letter that divides into every term.
   The terms of 4x² − 12x + 9 have no common factor other than 1, so there is nothing to take out.
2. Factorise the quadratic. It has three terms, so we use the "two numbers" method. Follow every line carefully.
   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 × (9) = 36. We need two numbers that multiply to 36 and add to −12 (the middle number).
   1 × 36 = 36,   add: 37 ✗
   −1 × −36 = 36,   add: −37 ✗
   2 × 18 = 36,   add: 20 ✗
   −2 × −18 = 36,   add: −20 ✗
   3 × 12 = 36,   add: 15 ✗
   −3 × −12 = 36,   add: −15 ✗
   4 × 9 = 36,   add: 13 ✗
   −4 × −9 = 36,   add: −13 ✗
   6 × 6 = 36,   add: 12 ✗
   −6 × −6 = 36,   add: −12 ✓ this pair works
   Split the middle term (−12x) into −6x and −6x: 4x² − 6x − 6x + 9.
   Factorise each pair separately. First pair: 4x² − 6x = 2x(2x − 3). Second pair: −6x + 9 = 3(−2x + 3).
   The two brackets (2x − 3) are the same. That is how we know it has worked. Write: 2x(2x − 3) + 3(−2x + 3).
   Take the common bracket out: (2x − 3) multiplied by (2x + 3).
   Result: (2x − 3)².
   Check by expanding: (2x − 3)² = 4x² − 12x + 9. ✓
3. Write the answer. The brackets are the factors of the quadratic. We have checked by expanding that they multiply back to the original.
   4x² − 12x + 9 = (2x − 3)²

Check. Expand the answer: (2x − 3)² gives 4x² − 12x + 9 again. ✓

Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong. And check for a common factor first: it makes the numbers smaller.

Key words. A factor of a term divides into it exactly. A common factor of several terms is a factor of all of them. To factorise an expression means to write it as a product, with brackets. It is the reverse of expanding.

Method. Factorising: (1) take out any common factor, (2) look at what is left: a difference of two squares or a quadratic with three terms, (3) check by expanding.

1. Check for a common factor first. Always check this first. A common factor is a number or letter that divides into every term.
   The terms of 2x² + x − 6 have no common factor other than 1, so there is nothing to take out.
2. Factorise the quadratic. It has three terms, so we use the "two numbers" method. Follow every line carefully.
   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 × (−6) = −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: 2x² − 3x + 4x − 6.
   Factorise each pair separately. First pair: 2x² − 3x = x(2x − 3). Second pair: 4x − 6 = 2(2x − 3).
   The two brackets (2x − 3) are the same. That is how we know it has worked. Write: x(2x − 3) + 2(2x − 3).
   Take the common bracket out: (2x − 3) multiplied by (x + 2).
   Result: (x + 2)(2x − 3).
   Check by expanding: (x + 2)(2x − 3) = 2x² + x − 6. ✓
3. Write the answer. The brackets are the factors of the quadratic. We have checked by expanding that they multiply back to the original.
   2x² + x − 6 = (x + 2)(2x − 3)

Check. Expand the answer: (x + 2)(2x − 3) gives 2x² + x − 6 again. ✓

Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong. And check for a common factor first: it makes the numbers smaller.

Key words. A factor of a term divides into it exactly. A common factor of several terms is a factor of all of them. To factorise an expression means to write it as a product, with brackets. It is the reverse of expanding.

Method. Factorising: (1) take out any common factor, (2) look at what is left: a difference of two squares or a quadratic with three terms, (3) check by expanding.

1. Check for a common factor first. Always check this first. A common factor is a number or letter that divides into every term.
   The terms of 3x² − 2x − 8 have no common factor other than 1, so there is nothing to take out.
2. Factorise the quadratic. It has three terms, so we use the "two numbers" method. Follow every line carefully.
   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 −2 (the middle number).
   1 × −24 = −24,   add: −23 ✗
   −1 × 24 = −24,   add: 23 ✗
   2 × −12 = −24,   add: −10 ✗
   −2 × 12 = −24,   add: 10 ✗
   3 × −8 = −24,   add: −5 ✗
   −3 × 8 = −24,   add: 5 ✗
   4 × −6 = −24,   add: −2 ✓ this pair works
   Split the middle term (−2x) 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 − 2x − 8. ✓
3. Write the answer. The brackets are the factors of the quadratic. We have checked by expanding that they multiply back to the original.
   3x² − 2x − 8 = (x − 2)(3x + 4)

Check. Expand the answer: (x − 2)(3x + 4) gives 3x² − 2x − 8 again. ✓

Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong. And check for a common factor first: it makes the numbers smaller.

Key words. A factor of a term divides into it exactly. A common factor of several terms is a factor of all of them. To factorise an expression means to write it as a product, with brackets. It is the reverse of expanding.

Method. Factorising: (1) take out any common factor, (2) look at what is left: a difference of two squares or a quadratic with three terms, (3) check by expanding.

1. Check for a common factor first. Always check this first. A common factor is a number or letter that divides into every term.
   The terms of 5x² + 13x − 6 have no common factor other than 1, so there is nothing to take out.
2. Factorise the quadratic. It has three terms, so we use the "two numbers" method. Follow every line carefully.
   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 × (−6) = −30. We need two numbers that multiply to −30 and add to 13 (the middle number).
   1 × −30 = −30,   add: −29 ✗
   −1 × 30 = −30,   add: 29 ✗
   2 × −15 = −30,   add: −13 ✗
   −2 × 15 = −30,   add: 13 ✓ this pair works
   Split the middle term (13x) into −2x and 15x: 5x² − 2x + 15x − 6.
   Factorise each pair separately. First pair: 5x² − 2x = x(5x − 2). Second pair: 15x − 6 = 3(5x − 2).
   The two brackets (5x − 2) are the same. That is how we know it has worked. Write: x(5x − 2) + 3(5x − 2).
   Take the common bracket out: (5x − 2) multiplied by (x + 3).
   Result: (x + 3)(5x − 2).
   Check by expanding: (x + 3)(5x − 2) = 5x² + 13x − 6. ✓
3. Write the answer. The brackets are the factors of the quadratic. We have checked by expanding that they multiply back to the original.
   5x² + 13x − 6 = (x + 3)(5x − 2)

Check. Expand the answer: (x + 3)(5x − 2) gives 5x² + 13x − 6 again. ✓

Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong. And check for a common factor first: it makes the numbers smaller.

Key words. A factor of a term divides into it exactly. A common factor of several terms is a factor of all of them. To factorise an expression means to write it as a product, with brackets. It is the reverse of expanding.

Method. Factorising: (1) take out any common factor, (2) look at what is left: a difference of two squares or a quadratic with three terms, (3) check by expanding.

1. Check for a common factor first. Always check this first. A common factor is a number or letter that divides into every term.
   The terms of 6x² + x − 2 have no common factor other than 1, so there is nothing to take out.
2. Factorise the quadratic. It has three terms, so we use the "two numbers" method. Follow every line carefully.
   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. ✓
3. Write the answer. The brackets are the factors of the quadratic. We have checked by expanding that they multiply back to the original.
   6x² + x − 2 = (2x − 1)(3x + 2)

Check. Expand the answer: (2x − 1)(3x + 2) gives 6x² + x − 2 again. ✓

Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong. And check for a common factor first: it makes the numbers smaller.

Key words. A factor of a term divides into it exactly. A common factor of several terms is a factor of all of them. To factorise an expression means to write it as a product, with brackets. It is the reverse of expanding.

Method. Factorising: (1) take out any common factor, (2) look at what is left: a difference of two squares or a quadratic with three terms, (3) check by expanding.

1. Check for a common factor first. Always check this first. A common factor is a number or letter that divides into every term.
   The terms of 4x² − 3x − 1 have no common factor other than 1, so there is nothing to take out.
2. Factorise the quadratic. It has three terms, so we use the "two numbers" method. Follow every line carefully.
   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 × (−1) = −4. We need two numbers that multiply to −4 and add to −3 (the middle number).
   1 × −4 = −4,   add: −3 ✓ this pair works
   Split the middle term (−3x) into 1x and −4x: 4x² + 1x − 4x − 1.
   Factorise each pair separately. First pair: 4x² + x = x(4x + 1). Second pair: −4x − 1 = −1(4x + 1).
   The two brackets (4x + 1) are the same. That is how we know it has worked. Write: x(4x + 1) − 1(4x + 1).
   Take the common bracket out: (4x + 1) multiplied by (x − 1).
   Result: (x − 1)(4x + 1).
   Check by expanding: (x − 1)(4x + 1) = 4x² − 3x − 1. ✓
3. Write the answer. The brackets are the factors of the quadratic. We have checked by expanding that they multiply back to the original.
   4x² − 3x − 1 = (x − 1)(4x + 1)

Check. Expand the answer: (x − 1)(4x + 1) gives 4x² − 3x − 1 again. ✓

Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong. And check for a common factor first: it makes the numbers smaller.

Key words. A factor of a term divides into it exactly. A common factor of several terms is a factor of all of them. To factorise an expression means to write it as a product, with brackets. It is the reverse of expanding.

Method. Factorising: (1) take out any common factor, (2) look at what is left: a difference of two squares or a quadratic with three terms, (3) check by expanding.

1. Check for a common factor first. Always check this first. A common factor is a number or letter that divides into every term.
   The terms of 6x² − 5x − 6 have no common factor other than 1, so there is nothing to take out.
2. Factorise the quadratic. It has three terms, so we use the "two numbers" method. Follow every line carefully.
   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 × (−6) = −36. We need two numbers that multiply to −36 and add to −5 (the middle number).
   1 × −36 = −36,   add: −35 ✗
   −1 × 36 = −36,   add: 35 ✗
   2 × −18 = −36,   add: −16 ✗
   −2 × 18 = −36,   add: 16 ✗
   3 × −12 = −36,   add: −9 ✗
   −3 × 12 = −36,   add: 9 ✗
   4 × −9 = −36,   add: −5 ✓ this pair works
   Split the middle term (−5x) into 4x and −9x: 6x² + 4x − 9x − 6.
   Factorise each pair separately. First pair: 6x² + 4x = 2x(3x + 2). Second pair: −9x − 6 = −3(3x + 2).
   The two brackets (3x + 2) are the same. That is how we know it has worked. Write: 2x(3x + 2) − 3(3x + 2).
   Take the common bracket out: (3x + 2) multiplied by (2x − 3).
   Result: (2x − 3)(3x + 2).
   Check by expanding: (2x − 3)(3x + 2) = 6x² − 5x − 6. ✓
3. Write the answer. The brackets are the factors of the quadratic. We have checked by expanding that they multiply back to the original.
   6x² − 5x − 6 = (2x − 3)(3x + 2)

Check. Expand the answer: (2x − 3)(3x + 2) gives 6x² − 5x − 6 again. ✓

Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong. And check for a common factor first: it makes the numbers smaller.

Key words. A factor of a term divides into it exactly. A common factor of several terms is a factor of all of them. To factorise an expression means to write it as a product, with brackets. It is the reverse of expanding.

Method. Factorising: (1) take out any common factor, (2) look at what is left: a difference of two squares or a quadratic with three terms, (3) check by expanding.

1. Write the terms and think: what could divide into every one? Factorising is the opposite of expanding: we take a common factor outside a bracket. A common factor is a number or letter that divides exactly into every term. We want the highest common factor, so that nothing is left to take out.
   The terms are: 12x², 10x, −8
2. Find the highest common factor of the numbers. List the factors of each number (all the numbers that divide into it exactly). The biggest number that is in every list is the number part of the common factor.
   Factors of 12: 1, 2, 3, 4, 6, 12
   Factors of 10: 1, 2, 5, 10
   Factors of 8: 1, 2, 4, 8
   The biggest number that is in every list is 2.
3. Find the letters that are in every term. A letter is part of the common factor only if it appears in every term. Then use the lowest power of it, because that is the most that divides into all of them (x divides into x² and into x, but x² does not divide into x).
   x: it is missing from at least one term (a missing letter has power 0), so it is not common.
4. Write the highest common factor. Multiply the number part and the common letters together.
   HCF = 2
5. Write the HCF outside a bracket. Inside the bracket put each term divided by the HCF. This is the reverse of expanding. Each division tells us what must be left inside the bracket.
   12x² ÷ 2 = 6x²
   10x ÷ 2 = 5x
   −8 ÷ 2 = −4
   So we can write 2(6x² + 5x − 4)
6. Look at the bracket. Can it be factorised again? After taking out the HCF, the bracket might be a difference of two squares or a quadratic that factorises. A question that says "factorise fully" needs every possible factor taken out.
   The bracket is (6x² + 5x − 4) = 6x² + 5x − 4.
7. Factorise the quadratic. It has three terms, so we use the "two numbers" method. Follow every line carefully.
   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 × (−4) = −24. We need two numbers that multiply to −24 and add to 5 (the middle number).
   1 × −24 = −24,   add: −23 ✗
   −1 × 24 = −24,   add: 23 ✗
   2 × −12 = −24,   add: −10 ✗
   −2 × 12 = −24,   add: 10 ✗
   3 × −8 = −24,   add: −5 ✗
   −3 × 8 = −24,   add: 5 ✓ this pair works
   Split the middle term (5x) into −3x and 8x: 6x² − 3x + 8x − 4.
   Factorise each pair separately. First pair: 6x² − 3x = 3x(2x − 1). Second pair: 8x − 4 = 4(2x − 1).
   The two brackets (2x − 1) are the same. That is how we know it has worked. Write: 3x(2x − 1) + 4(2x − 1).
   Take the common bracket out: (2x − 1) multiplied by (3x + 4).
   Result: (2x − 1)(3x + 4).
   Check by expanding: (2x − 1)(3x + 4) = 6x² + 5x − 4. ✓
8. Write the full answer. Do not forget the common factor you took out at the start. It stays in front of the brackets.
   12x² + 10x − 8 = 2(2x − 1)(3x + 4)

Check. Expand the answer: 2(2x − 1)(3x + 4) gives 12x² + 10x − 8 again. ✓

Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong. And check for a common factor first: it makes the numbers smaller.

Key words. A factor of a term divides into it exactly. A common factor of several terms is a factor of all of them. To factorise an expression means to write it as a product, with brackets. It is the reverse of expanding.

Method. Factorising: (1) take out any common factor, (2) look at what is left: a difference of two squares or a quadratic with three terms, (3) check by expanding.

1. Write the terms and think: what could divide into every one? Factorising is the opposite of expanding: we take a common factor outside a bracket. A common factor is a number or letter that divides exactly into every term. We want the highest common factor, so that nothing is left to take out.
   The terms are: 6x², 14x, 4
2. Find the highest common factor of the numbers. List the factors of each number (all the numbers that divide into it exactly). The biggest number that is in every list is the number part of the common factor.
   Factors of 6: 1, 2, 3, 6
   Factors of 14: 1, 2, 7, 14
   Factors of 4: 1, 2, 4
   The biggest number that is in every list is 2.
3. Find the letters that are in every term. A letter is part of the common factor only if it appears in every term. Then use the lowest power of it, because that is the most that divides into all of them (x divides into x² and into x, but x² does not divide into x).
   x: it is missing from at least one term (a missing letter has power 0), so it is not common.
4. Write the highest common factor. Multiply the number part and the common letters together.
   HCF = 2
5. Write the HCF outside a bracket. Inside the bracket put each term divided by the HCF. This is the reverse of expanding. Each division tells us what must be left inside the bracket.
   6x² ÷ 2 = 3x²
   14x ÷ 2 = 7x
   4 ÷ 2 = 2
   So we can write 2(3x² + 7x + 2)
6. Look at the bracket. Can it be factorised again? After taking out the HCF, the bracket might be a difference of two squares or a quadratic that factorises. A question that says "factorise fully" needs every possible factor taken out.
   The bracket is (3x² + 7x + 2) = 3x² + 7x + 2.
7. Factorise the quadratic. It has three terms, so we use the "two numbers" method. Follow every line carefully.
   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 × (2) = 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: 3x² + 1x + 6x + 2.
   Factorise each pair separately. First pair: 3x² + x = x(3x + 1). Second pair: 6x + 2 = 2(3x + 1).
   The two brackets (3x + 1) are the same. That is how we know it has worked. Write: x(3x + 1) + 2(3x + 1).
   Take the common bracket out: (3x + 1) multiplied by (x + 2).
   Result: (x + 2)(3x + 1).
   Check by expanding: (x + 2)(3x + 1) = 3x² + 7x + 2. ✓
8. Write the full answer. Do not forget the common factor you took out at the start. It stays in front of the brackets.
   6x² + 14x + 4 = 2(x + 2)(3x + 1)

Check. Expand the answer: 2(x + 2)(3x + 1) gives 6x² + 14x + 4 again. ✓

Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong. And check for a common factor first: it makes the numbers smaller.

Key words. A factor of a term divides into it exactly. A common factor of several terms is a factor of all of them. To factorise an expression means to write it as a product, with brackets. It is the reverse of expanding.

Method. Factorising: (1) take out any common factor, (2) look at what is left: a difference of two squares or a quadratic with three terms, (3) check by expanding.

1. Write the terms and think: what could divide into every one? Factorising is the opposite of expanding: we take a common factor outside a bracket. A common factor is a number or letter that divides exactly into every term. We want the highest common factor, so that nothing is left to take out.
   The terms are: 9x², −6x, −24
2. Find the highest common factor of the numbers. List the factors of each number (all the numbers that divide into it exactly). The biggest number that is in every list is the number part of the common factor.
   Factors of 9: 1, 3, 9
   Factors of 6: 1, 2, 3, 6
   Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
   The biggest number that is in every list is 3.
3. Find the letters that are in every term. A letter is part of the common factor only if it appears in every term. Then use the lowest power of it, because that is the most that divides into all of them (x divides into x² and into x, but x² does not divide into x).
   x: it is missing from at least one term (a missing letter has power 0), so it is not common.
4. Write the highest common factor. Multiply the number part and the common letters together.
   HCF = 3
5. Write the HCF outside a bracket. Inside the bracket put each term divided by the HCF. This is the reverse of expanding. Each division tells us what must be left inside the bracket.
   9x² ÷ 3 = 3x²
   −6x ÷ 3 = −2x
   −24 ÷ 3 = −8
   So we can write 3(3x² − 2x − 8)
6. Look at the bracket. Can it be factorised again? After taking out the HCF, the bracket might be a difference of two squares or a quadratic that factorises. A question that says "factorise fully" needs every possible factor taken out.
   The bracket is (3x² − 2x − 8) = 3x² − 2x − 8.
7. Factorise the quadratic. It has three terms, so we use the "two numbers" method. Follow every line carefully.
   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 −2 (the middle number).
   1 × −24 = −24,   add: −23 ✗
   −1 × 24 = −24,   add: 23 ✗
   2 × −12 = −24,   add: −10 ✗
   −2 × 12 = −24,   add: 10 ✗
   3 × −8 = −24,   add: −5 ✗
   −3 × 8 = −24,   add: 5 ✗
   4 × −6 = −24,   add: −2 ✓ this pair works
   Split the middle term (−2x) 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 − 2x − 8. ✓
8. Write the full answer. Do not forget the common factor you took out at the start. It stays in front of the brackets.
   9x² − 6x − 24 = 3(x − 2)(3x + 4)

Check. Expand the answer: 3(x − 2)(3x + 4) gives 9x² − 6x − 24 again. ✓

Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong. And check for a common factor first: it makes the numbers smaller.

Key words. A factor of a term divides into it exactly. A common factor of several terms is a factor of all of them. To factorise an expression means to write it as a product, with brackets. It is the reverse of expanding.

Method. Factorising: (1) take out any common factor, (2) look at what is left: a difference of two squares or a quadratic with three terms, (3) check by expanding.

1. Write the terms and think: what could divide into every one? Factorising is the opposite of expanding: we take a common factor outside a bracket. A common factor is a number or letter that divides exactly into every term. We want the highest common factor, so that nothing is left to take out.
   The terms are: 10x², −5x, −5
2. Find the highest common factor of the numbers. List the factors of each number (all the numbers that divide into it exactly). The biggest number that is in every list is the number part of the common factor.
   Factors of 10: 1, 2, 5, 10
   Factors of 5: 1, 5
   Factors of 5: 1, 5
   The biggest number that is in every list is 5.
3. Find the letters that are in every term. A letter is part of the common factor only if it appears in every term. Then use the lowest power of it, because that is the most that divides into all of them (x divides into x² and into x, but x² does not divide into x).
   x: it is missing from at least one term (a missing letter has power 0), so it is not common.
4. Write the highest common factor. Multiply the number part and the common letters together.
   HCF = 5
5. Write the HCF outside a bracket. Inside the bracket put each term divided by the HCF. This is the reverse of expanding. Each division tells us what must be left inside the bracket.
   10x² ÷ 5 = 2x²
   −5x ÷ 5 = −x
   −5 ÷ 5 = −1
   So we can write 5(2x² − x − 1)
6. Look at the bracket. Can it be factorised again? After taking out the HCF, the bracket might be a difference of two squares or a quadratic that factorises. A question that says "factorise fully" needs every possible factor taken out.
   The bracket is (2x² − x − 1) = 2x² − x − 1.
7. Factorise the quadratic. It has three terms, so we use the "two numbers" method. Follow every line carefully.
   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 × (−1) = −2. We need two numbers that multiply to −2 and add to −1 (the middle number).
   1 × −2 = −2,   add: −1 ✓ this pair works
   Split the middle term (−1x) into 1x and −2x: 2x² + 1x − 2x − 1.
   Factorise each pair separately. First pair: 2x² + x = x(2x + 1). Second pair: −2x − 1 = −1(2x + 1).
   The two brackets (2x + 1) are the same. That is how we know it has worked. Write: x(2x + 1) − 1(2x + 1).
   Take the common bracket out: (2x + 1) multiplied by (x − 1).
   Result: (x − 1)(2x + 1).
   Check by expanding: (x − 1)(2x + 1) = 2x² − x − 1. ✓
8. Write the full answer. Do not forget the common factor you took out at the start. It stays in front of the brackets.
   10x² − 5x − 5 = 5(x − 1)(2x + 1)

Check. Expand the answer: 5(x − 1)(2x + 1) gives 10x² − 5x − 5 again. ✓

Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong. And check for a common factor first: it makes the numbers smaller.

Key words. A factor of a term divides into it exactly. A common factor of several terms is a factor of all of them. To factorise an expression means to write it as a product, with brackets. It is the reverse of expanding.

Method. Factorising: (1) take out any common factor, (2) look at what is left: a difference of two squares or a quadratic with three terms, (3) check by expanding.

1. Check for a common factor first. Always check this first. A common factor is a number or letter that divides into every term.
   The terms of 9x² − 4 have no common factor other than 1, so there is nothing to take out.
2. Look at the expression. Count the terms and check for squares. A difference of two squares has exactly two terms, both are perfect squares (something multiplied by itself), and there is a minus between them.
   9x² − 4:   9x² is a square and 4 is a square, with a minus between them. ✓
3. Remember the rule. It comes from expanding (a − b)(a + b) = a² + ab − ab − b² = a² − b². The two middle terms cancel. Going backwards:
   a² − b² = (a − b)(a + b)
4. Find a and b by square rooting each term. The square root undoes the square. Take the square root of the number, and halve the power of each letter.
   9x² = (3x)², so a = 3x
   4 = (2)², so b = 2
5. Put a and b into the rule. One bracket has a minus, the other has a plus.
   (3x − 2)(3x + 2)

Check. Expand to check. (3x − 2)(3x + 2): 3x × 3x = 9x²; 3x × 2 = 6x and (−2) × 3x = −6x (these two cancel); (−2) × 2 = −4. Total: 9x² − 4. ✓

Watch out. It only works with a minus between two squares. x² + 9 cannot be factorised like this. Square root the number and the letter: 25x² gives 5x, not 25x.

Key words. A factor of a term divides into it exactly. A common factor of several terms is a factor of all of them. To factorise an expression means to write it as a product, with brackets. It is the reverse of expanding.

Method. Always look for a common factor first. Take out the highest common factor, then see whether the bracket can be factorised further.

1. Write the terms and think: what could divide into every one? Factorising is the opposite of expanding: we take a common factor outside a bracket. A common factor is a number or letter that divides exactly into every term. We want the highest common factor, so that nothing is left to take out.
   The terms are: 4x², 8x
2. Find the highest common factor of the numbers. List the factors of each number (all the numbers that divide into it exactly). The biggest number that is in every list is the number part of the common factor.
   Factors of 4: 1, 2, 4
   Factors of 8: 1, 2, 4, 8
   The biggest number that is in every list is 4.
3. Find the letters that are in every term. A letter is part of the common factor only if it appears in every term. Then use the lowest power of it, because that is the most that divides into all of them (x divides into x² and into x, but x² does not divide into x).
   x: the powers in the terms are 2, 1. It is in every term and the lowest power is 1, so x is common.
4. Write the highest common factor. Multiply the number part and the common letters together.
   HCF = 4x
5. Write the HCF outside a bracket. Inside the bracket put each term divided by the HCF. This is the reverse of expanding. Each division tells us what must be left inside the bracket.
   4x² ÷ 4x = x
   8x ÷ 4x = 2
   So we can write 4x(x + 2)
   The bracket has no common factor left and cannot be factorised again, so this is the answer.

Check. Expand: 4x(x + 2) = 4x² + 8x. ✓

Watch out. Take out the highest common factor. If you take out only 2 from 6x + 12, the bracket (3x + 6) still has a common factor, so it is not fully factorised. And if a term is the common factor itself, the bracket gets a 1, not a blank: 3x + 3 = 3(x + 1).

Key words. A factor of a term divides into it exactly. A common factor of several terms is a factor of all of them. To factorise an expression means to write it as a product, with brackets. It is the reverse of expanding.

Method. Factorising: (1) take out any common factor, (2) look at what is left: a difference of two squares or a quadratic with three terms, (3) check by expanding.

1. Check for a common factor first. Always check this first. A common factor is a number or letter that divides into every term.
   The terms of 2x² + 11x + 5 have no common factor other than 1, so there is nothing to take out.
2. Factorise the quadratic. It has three terms, so we use the "two numbers" method. Follow every line carefully.
   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 × (5) = 10. We need two numbers that multiply to 10 and add to 11 (the middle number).
   1 × 10 = 10,   add: 11 ✓ this pair works
   Split the middle term (11x) into 1x and 10x: 2x² + 1x + 10x + 5.
   Factorise each pair separately. First pair: 2x² + x = x(2x + 1). Second pair: 10x + 5 = 5(2x + 1).
   The two brackets (2x + 1) are the same. That is how we know it has worked. Write: x(2x + 1) + 5(2x + 1).
   Take the common bracket out: (2x + 1) multiplied by (x + 5).
   Result: (x + 5)(2x + 1).
   Check by expanding: (x + 5)(2x + 1) = 2x² + 11x + 5. ✓
3. Write the answer. The brackets are the factors of the quadratic. We have checked by expanding that they multiply back to the original.
   2x² + 11x + 5 = (x + 5)(2x + 1)

Check. Expand the answer: (x + 5)(2x + 1) gives 2x² + 11x + 5 again. ✓

Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong. And check for a common factor first: it makes the numbers smaller.

Key words. A factor of a term divides into it exactly. A common factor of several terms is a factor of all of them. To factorise an expression means to write it as a product, with brackets. It is the reverse of expanding.

Method. Factorising: (1) take out any common factor, (2) look at what is left: a difference of two squares or a quadratic with three terms, (3) check by expanding.

1. Check for a common factor first. Always check this first. A common factor is a number or letter that divides into every term.
   The terms of x² − 3x − 10 have no common factor other than 1, so there is nothing to take out.
2. Factorise the quadratic. It has three terms, so we use the "two numbers" method. Follow every line carefully.
   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. ✓
3. Write the answer. The brackets are the factors of the quadratic. We have checked by expanding that they multiply back to the original.
   x² − 3x − 10 = (x − 5)(x + 2)

Check. Expand the answer: (x − 5)(x + 2) gives x² − 3x − 10 again. ✓

Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong. And check for a common factor first: it makes the numbers smaller.

Key words. A factor of a term divides into it exactly. A common factor of several terms is a factor of all of them. To factorise an expression means to write it as a product, with brackets. It is the reverse of expanding.

Method. Factorising: (1) take out any common factor, (2) look at what is left: a difference of two squares or a quadratic with three terms, (3) check by expanding.

1. Check for a common factor first. Always check this first. A common factor is a number or letter that divides into every term.
   The terms of 6x² − 7x − 3 have no common factor other than 1, so there is nothing to take out.
2. Factorise the quadratic. It has three terms, so we use the "two numbers" method. Follow every line carefully.
   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 × (−3) = −18. We need two numbers that multiply to −18 and add to −7 (the middle number).
   1 × −18 = −18,   add: −17 ✗
   −1 × 18 = −18,   add: 17 ✗
   2 × −9 = −18,   add: −7 ✓ this pair works
   Split the middle term (−7x) into 2x and −9x: 6x² + 2x − 9x − 3.
   Factorise each pair separately. First pair: 6x² + 2x = 2x(3x + 1). Second pair: −9x − 3 = −3(3x + 1).
   The two brackets (3x + 1) are the same. That is how we know it has worked. Write: 2x(3x + 1) − 3(3x + 1).
   Take the common bracket out: (3x + 1) multiplied by (2x − 3).
   Result: (2x − 3)(3x + 1).
   Check by expanding: (2x − 3)(3x + 1) = 6x² − 7x − 3. ✓
3. Write the answer. The brackets are the factors of the quadratic. We have checked by expanding that they multiply back to the original.
   6x² − 7x − 3 = (2x − 3)(3x + 1)

Check. Expand the answer: (2x − 3)(3x + 1) gives 6x² − 7x − 3 again. ✓

Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong. And check for a common factor first: it makes the numbers smaller.

Key words. A factor of a term divides into it exactly. A common factor of several terms is a factor of all of them. To factorise an expression means to write it as a product, with brackets. It is the reverse of expanding.

Method. Factorising: (1) take out any common factor, (2) look at what is left: a difference of two squares or a quadratic with three terms, (3) check by expanding.

1. Check for a common factor first. Always check this first. A common factor is a number or letter that divides into every term.
   The terms of 2y² + 5y + 2 have no common factor other than 1, so there is nothing to take out.
2. Factorise the quadratic. It has three terms, so we use the "two numbers" method. Follow every line carefully.
   Spot the pattern. A quadratic where the number in front of y² is 2, not 1. We use the split the middle term method.
   Multiply the first and last numbers: 2 × (2) = 4. We need two numbers that multiply to 4 and add to 5 (the middle number).
   1 × 4 = 4,   add: 5 ✓ this pair works
   Split the middle term (5y) into 1y and 4y: 2y² + 1y + 4y + 2.
   Factorise each pair separately. First pair: 2y² + y = y(2y + 1). Second pair: 4y + 2 = 2(2y + 1).
   The two brackets (2y + 1) are the same. That is how we know it has worked. Write: y(2y + 1) + 2(2y + 1).
   Take the common bracket out: (2y + 1) multiplied by (y + 2).
   Result: (y + 2)(2y + 1).
   Check by expanding: (y + 2)(2y + 1) = 2y² + 5y + 2. ✓
3. Write the answer. The brackets are the factors of the quadratic. We have checked by expanding that they multiply back to the original.
   2y² + 5y + 2 = (y + 2)(2y + 1)

Check. Expand the answer: (y + 2)(2y + 1) gives 2y² + 5y + 2 again. ✓

Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong. And check for a common factor first: it makes the numbers smaller.

Key words. A factor of a term divides into it exactly. A common factor of several terms is a factor of all of them. To factorise an expression means to write it as a product, with brackets. It is the reverse of expanding.

Method. Factorising: (1) take out any common factor, (2) look at what is left: a difference of two squares or a quadratic with three terms, (3) check by expanding.

1. Check for a common factor first. Always check this first. A common factor is a number or letter that divides into every term.
   The terms of 3p² − 4p − 4 have no common factor other than 1, so there is nothing to take out.
2. Factorise the quadratic. It has three terms, so we use the "two numbers" method. Follow every line carefully.
   Spot the pattern. A quadratic where the number in front of p² is 3, not 1. We use the split the middle term method.
   Multiply the first and last numbers: 3 × (−4) = −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 (−4p) into 2p and −6p: 3p² + 2p − 6p − 4.
   Factorise each pair separately. First pair: 3p² + 2p = p(3p + 2). Second pair: −6p − 4 = −2(3p + 2).
   The two brackets (3p + 2) are the same. That is how we know it has worked. Write: p(3p + 2) − 2(3p + 2).
   Take the common bracket out: (3p + 2) multiplied by (p − 2).
   Result: (p − 2)(3p + 2).
   Check by expanding: (p − 2)(3p + 2) = 3p² − 4p − 4. ✓
3. Write the answer. The brackets are the factors of the quadratic. We have checked by expanding that they multiply back to the original.
   3p² − 4p − 4 = (p − 2)(3p + 2)

Check. Expand the answer: (p − 2)(3p + 2) gives 3p² − 4p − 4 again. ✓

Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong. And check for a common factor first: it makes the numbers smaller.

Key words. A factor of a term divides into it exactly. A common factor of several terms is a factor of all of them. To factorise an expression means to write it as a product, with brackets. It is the reverse of expanding.

Method. Factorising: (1) take out any common factor, (2) look at what is left: a difference of two squares or a quadratic with three terms, (3) check by expanding.

1. Check for a common factor first. Always check this first. A common factor is a number or letter that divides into every term.
   The terms of 6m² − 17m + 12 have no common factor other than 1, so there is nothing to take out.
2. Factorise the quadratic. It has three terms, so we use the "two numbers" method. Follow every line carefully.
   Spot the pattern. A quadratic where the number in front of m² is 6, not 1. We use the split the middle term method.
   Multiply the first and last numbers: 6 × (12) = 72. We need two numbers that multiply to 72 and add to −17 (the middle number).
   1 × 72 = 72,   add: 73 ✗
   −1 × −72 = 72,   add: −73 ✗
   2 × 36 = 72,   add: 38 ✗
   −2 × −36 = 72,   add: −38 ✗
   3 × 24 = 72,   add: 27 ✗
   −3 × −24 = 72,   add: −27 ✗
   4 × 18 = 72,   add: 22 ✗
   −4 × −18 = 72,   add: −22 ✗
   6 × 12 = 72,   add: 18 ✗
   −6 × −12 = 72,   add: −18 ✗
   8 × 9 = 72,   add: 17 ✗
   −8 × −9 = 72,   add: −17 ✓ this pair works
   Split the middle term (−17m) into −8m and −9m: 6m² − 8m − 9m + 12.
   Factorise each pair separately. First pair: 6m² − 8m = 2m(3m − 4). Second pair: −9m + 12 = 3(−3m + 4).
   The two brackets (3m − 4) are the same. That is how we know it has worked. Write: 2m(3m − 4) + 3(−3m + 4).
   Take the common bracket out: (3m − 4) multiplied by (2m + 3).
   Result: (2m − 3)(3m − 4).
   Check by expanding: (2m − 3)(3m − 4) = 6m² − 17m + 12. ✓
3. Write the answer. The brackets are the factors of the quadratic. We have checked by expanding that they multiply back to the original.
   6m² − 17m + 12 = (2m − 3)(3m − 4)

Check. Expand the answer: (2m − 3)(3m − 4) gives 6m² − 17m + 12 again. ✓

Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong. And check for a common factor first: it makes the numbers smaller.

Key words. A factor of a term divides into it exactly. A common factor of several terms is a factor of all of them. To factorise an expression means to write it as a product, with brackets. It is the reverse of expanding.

Method. Factorising: (1) take out any common factor, (2) look at what is left: a difference of two squares or a quadratic with three terms, (3) check by expanding.

1. Check for a common factor first. Always check this first. A common factor is a number or letter that divides into every term.
   The terms of 6x² + 7x + 2 have no common factor other than 1, so there is nothing to take out.
2. Factorise the quadratic. It has three terms, so we use the "two numbers" method. Follow every line carefully.
   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 7 (the middle number).
   1 × 12 = 12,   add: 13 ✗
   −1 × −12 = 12,   add: −13 ✗
   2 × 6 = 12,   add: 8 ✗
   −2 × −6 = 12,   add: −8 ✗
   3 × 4 = 12,   add: 7 ✓ this pair works
   Split the middle term (7x) 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 + 7x + 2. ✓
3. Write the answer. The brackets are the factors of the quadratic. We have checked by expanding that they multiply back to the original.
   6x² + 7x + 2 = (2x + 1)(3x + 2)

Check. Expand the answer: (2x + 1)(3x + 2) gives 6x² + 7x + 2 again. ✓

Watch out. Always check by expanding. If you do not get the original expression back, the factorising is wrong. And check for a common factor first: it makes the numbers smaller.

Key words. A term is one part of an expression, together with the sign in front of it (for example 3x or −5). Like terms have exactly the same letters with exactly the same powers (3x and 5x are like terms, 3x and 5x² are not). The coefficient is the number in front of the letters.

Method. Collect like terms: group the terms that have the same letters and powers, then add or subtract their coefficients (the numbers in front).

1. Write the terms out one by one, each with the sign in front of it. A term is one part of the expression. The sign in front of a term belongs to that term, so in 5x − 2y the terms are +5x and −2y. (A term with no sign in front is positive.)
   +3x, +2, +2x, +1, +3x, +2, +2x, +1
2. Sort the terms into groups of like terms. Like terms have the same letters with the same powers, so they are the same kind of thing. We can only add the same kind of thing: 3 apples + 2 apples = 5 apples, but 3 apples + 2 pears cannot be made into one number. Remember that x and x² are different kinds of term, and so are x and xy.
   x terms: +3x, +2x, +3x, +2x
      add the coefficients: 3 + 2 + 3 + 2 = 10  so they give 10x
   number terms: +2, +1, +2, +1
      add the coefficients: 2 + 1 + 2 + 1 = 6  so they give 6
3. Write the answer. Join the results of the groups. Put the term with the highest power first, and the number last.
   10x + 6

Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 36 and the answer gives 36. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. The sign in front of a term goes with it. In 7x − 3x you are taking away 3 lots of x. Also, a letter on its own, such as x, has a coefficient of 1 (and −x has −1).