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KS3 Maths · Year 7Chapter 5 · Simplifying, expanding and factorising
Year 7 · Chapter 5

Simplifying, expanding and factorising

By the end of this chapter you will be able to

  • Collect like terms to simplify an expression, and know which terms are like terms
  • Expand a single bracket, including with a negative number or a letter outside the bracket
  • Expand the product of two brackets, using FOIL, and collect the terms
  • Factorise an expression by taking out the highest common factor
  • Multiply and divide terms that have numbers and powers of letters

You should already know

  • Substitution and the words of algebra (Chapter 4)
  • Multiplication facts and the order of operations (Chapter 1)
  • Multiplication of brackets with numbers, such as 3(x + 2) (Year 6, Chapter 4)

Where this leads

  • Solving equations with brackets (Chapter 6)
  • Quadratic expressions and factorising quadratics (Year 8)

National Curriculum for England, Key Stage 3 mathematics, Year 7. Content: A4 simplify and manipulate expressions while keeping equivalence: collect like terms, multiply a single term over a bracket, take out common factors, expand products of two or more binomials.

Exercise P  Do you remember?

Non-calculator
1
FSimplify 3a + 4a.
2
FWork out 4 × (2 + 5).
3
FWork out 3 × 4 + 2 × 5.
4
FWhat is 5 squared?
5
FWork out −3 + 7.
5.1

Collecting like terms

Year 7
A4
Video lesson: Collecting like termsComing soon
A triangle has sides of 3x cm, 2x + 1 cm and x + 4 cm. Its perimeter is the total of the three sides. To write the perimeter as simply as possible, we collect the terms that are alike.

Like terms have the same letter raised to the same power: 3x and 5x are like terms, and so are 2y2 and −y2. Numbers on their own are like terms with each other. 3x and 3y are not like terms, and neither are x and x2.

To simplify, add or subtract the numbers in front of the like terms, and keep the letter and power the same.

Key ideas

  • Like terms have the same letter and the same power.
  • Add or subtract the numbers in front; keep the letter.
  • A letter on its own has a coefficient of 1: x = 1x.
  • Unlike terms, such as 3a and 2b, cannot be combined.
Worked example 5.1.1
Simplify 3a + 2b + 5a − b.

Method. Group the like terms together, then add or subtract their numbers. Keep the letter and the power the same.

  1. Find the a terms. The like terms 3a and 5a both have a.
    3a + 5a
  2. Find the b terms. The like terms 2b and −b both have b.
    2b − b
  3. Add the like terms. Add the numbers in front, and keep the letters.
    8a + b
  4. Write the answer. Read the answer back in its simplest form.
    8a + b

Check. Test with a = 1 and b = 2: 3 + 4 + 5 − 2 = 10, and 8 + 2 = 10. ✓

Watch out. The b on its own has a coefficient of 1, so write b, not 1b.

Worked example 5.1.2
Simplify 4x + 3 − 2x + 7.

Method. Group the like terms together, then add or subtract their numbers. Keep the letter and the power the same.

  1. Find the x terms. 4x and −2x are like terms.
    4x − 2x
  2. Find the number terms. 3 and 7 are like terms, because they have no letter.
    3 + 7
  3. Add the like terms. Add the x terms and add the numbers.
    2x + 10
  4. Write the answer. Read the answer back in its simplest form.
    2x + 10

Check. With x = 1: 4 + 3 − 2 + 7 = 12, and 2 + 10 = 12. ✓

Watch out. Do not combine the x and the number: 2x + 10 cannot be simplified to 12x.

Worked example 5.1.3
Simplify 2p2 + 3p − p2 + 5p.

Method. Group the like terms together, then add or subtract their numbers. Keep the letter and the power the same.

  1. Find the p2 terms. 2p2 and −p2 are like terms.
    2p2 − p2 = p2
  2. Find the p terms. 3p and 5p are like terms.
    3p + 5p = 8p
  3. Write the answer. The p2 and p terms are different, so they stay separate.
    p2 + 8p

Check. With p = 2: 8 + 6 − 4 + 10 = 20, and p2 + 8p = 4 + 16 = 20. ✓

Watch out. p2 and p are not like terms: they must not be added together.

Watch out

Watch out: 3x + 2y is not 5xy, and 3a + 4 is not 7a. Only terms with the same letter and power are like terms.

Exam tip

Grown-up tip. Ask the pupil to underline each like term with a different colour, to check nothing is missed.

Exercise 5A  Collecting like terms

Non-calculator
1
FSimplify 3a + 4a.
2
FSimplify 5x + 2x − x.
3
FSimplify 4p + 3q + 2p.
4
FSimplify 7 + 2y + 5.
5
FSimplify 9t − 4t + 3.
6
FSimplify 2a + 3b − a + 4b.
7
FSimplify 6m − 2n − 4m + 5n.
8
FSimplify x2 + 3x2.
9
FSimplify 5y2 − 2y + y2 + y.
10
FSimplify: (a) 3x + 2 + 4x (b) 10 − 2k − 3 (c) 4ab + 3ab
(a)(a) 3x + 2 + 4x
(b)(b) 10 − 2k − 3
(c)(c) 4ab + 3ab
11
PSA triangle has sides 3x, 2x + 1 and x + 4. Write its perimeter in its simplest form.
12
RSam says that 3a + 2b = 5ab. Is he right? Explain.
5.2

Expanding a single bracket

Year 7
A4
Video lesson: Expanding a single bracketComing soon
A rectangular garden is 3 m wide. Its length is x + 6 metres. The area is 3 × (x + 6), and multiplying the 3 by each part of the bracket gives an expression with no brackets.

Expanding a bracket means multiplying the term outside by every term inside. For example, 3(x + 4) = 3 × x + 3 × 4 = 3x + 12.

The sign outside the bracket is part of the multiplier. −2(x + 3) = −2x − 6, because −2 × 3 = −6. A letter outside the bracket multiplies too: x(x − 5) = x2 − 5x.

Key ideas

  • Multiply the term outside by every term inside the bracket.
  • Keep the sign of each term. A negative times a positive is negative.
  • x × x = x2, so x(x + 3) = x2 + 3x.
  • After expanding, collect any like terms.
Worked example 5.2.1
Expand 3(x + 4).

Method. Multiply the term outside the bracket by every term inside the bracket. Keep the signs.

  1. Multiply the outside term by the first term inside. Each term inside the bracket is multiplied, so none is missed.
    3 × x = 3x
  2. Multiply by the other terms inside. Take each term in turn; a negative term gives a negative product.
    3 × 4 = 12
  3. Write the answer. Collect any like terms, if there are any.
    3x + 12

Check. Test with x = 1 on both sides to check. ✓

Watch out. The outside term must multiply every term inside. Do not multiply only the first term.

Worked example 5.2.2
Expand −2(x + 3).

Method. Multiply the term outside the bracket by every term inside the bracket. Keep the signs.

  1. Multiply the outside term by the first term inside. Each term inside the bracket is multiplied, so none is missed.
    (−2) × x = −2x
  2. Multiply by the other terms inside. Take each term in turn; a negative term gives a negative product.
    (−2) × 3 = −6
  3. Write the answer. Collect any like terms, if there are any.
    −2x − 6

Check. Test with x = 1 on both sides to check. ✓

Watch out. The outside term must multiply every term inside. Do not multiply only the first term.

Worked example 5.2.3
Expand x(x − 5).

Method. Multiply the term outside the bracket by every term inside the bracket. Keep the signs.

  1. Multiply the outside term by the first term inside. Each term inside the bracket is multiplied, so none is missed.
    x × x = x2
  2. Multiply by the other terms inside. Take each term in turn; a negative term gives a negative product.
    x × (−5) = −5x
  3. Write the answer. Collect any like terms, if there are any.
    x2 − 5x

Check. Test with x = 1 on both sides to check. ✓

Watch out. The outside term must multiply every term inside. Do not multiply only the first term.

Watch out

Watch out: 3(x + 4) is 3x + 12, not 3x + 4. The 3 must multiply the 4 as well. Also −2(x + 3) is not −2x + 3.

Exam tip

Grown-up tip. Ask the pupil to draw arrows from the outside term to each term inside, so that no term is missed.

Exercise 5B  Expanding a single bracket

Non-calculator
1
FExpand 2(x + 5).
2
FExpand 4(y − 3).
3
FExpand 5(2a + 1), writing a for x.
4
FExpand 3(4 − x).
5
FExpand −3(x + 2).
6
FExpand x(x + 7).
7
FExpand 2x(3x − 1).
8
FExpand −x(x − 4).
9
FExpand and simplify 3(x + 4) + 2(x − 1).
10
FExpand: (a) 6(2 + x) (b) 4(3x − 2) (c) −(x + 5)
(a)(a) 6(2 + x)
(b)(b) 4(3x − 2)
(c)(c) −(x + 5)
11
PSA rectangle has length x + 6 cm and width 3 cm. Write an expression for its area.
12
RSam says that 2(x + 3) = 2x + 3. Is he right? Explain.
5.3

Expanding two brackets

Year 7
A4
Video lesson: Expanding two bracketsComing soon
A poster is (x + 2) cm wide and (x + 5) cm tall. Its area is (x + 2)(x + 5) cm2. Every term in the first bracket must be multiplied by every term in the second bracket.

To expand two brackets, use FOIL: multiply the First terms, the Outer terms, the Inner terms and the Last terms. Then collect the like terms. For example, (x + 2)(x + 5) = x2 + 5x + 2x + 10 = x2 + 7x + 10.

Key ideas

  • Multiply every term in one bracket by every term in the other: four products.
  • FOIL: First, Outside, Inside, Last.
  • The two x terms are like terms, so add them.
  • Test your answer by putting x = 1 into both forms.
Worked example 5.3.1
Expand (x + 2)(x + 5).

Method. Multiply every term in the first bracket by every term in the second bracket. Use FOIL: First, Outside, Inside, Last.

  1. First: multiply the first terms. The first term in each bracket is multiplied first.
    x × x = x2
  2. Outside: multiply the outer terms. The first term of the first bracket times the second term of the second.
    x × 5 = 5x
  3. Inside: multiply the inner terms. The second term of the first bracket times the first term of the second.
    2 × x = 2x
  4. Last: multiply the last terms. The two numbers at the end are multiplied.
    2 × 5 = 10
  5. Collect the like terms. Add the two x terms together and leave the x squared and the number as they are.
    x2 + 7x + 10

Check. Test with x = 1 in the brackets and in the answer: the two give the same value. ✓

Watch out. The two x terms are like terms, so combine them. Do not write x2 as 2x.

Worked example 5.3.2
Expand (x − 3)(x + 4).

Method. Multiply every term in the first bracket by every term in the second bracket. Use FOIL: First, Outside, Inside, Last.

  1. First: multiply the first terms. The first term in each bracket is multiplied first.
    x × x = x2
  2. Outside: multiply the outer terms. The first term of the first bracket times the second term of the second.
    x × 4 = 4x
  3. Inside: multiply the inner terms. The second term of the first bracket times the first term of the second.
    (−3) × x = −3x
  4. Last: multiply the last terms. The two numbers at the end are multiplied.
    (−3) × 4 = −12
  5. Collect the like terms. Add the two x terms together and leave the x squared and the number as they are.
    x2 + x − 12

Check. Test with x = 1 in the brackets and in the answer: the two give the same value. ✓

Watch out. The two x terms are like terms, so combine them. Do not write x2 as 2x.

Worked example 5.3.3
Expand (2x + 1)(x − 3).

Method. Multiply every term in the first bracket by every term in the second bracket. Use FOIL: First, Outside, Inside, Last.

  1. First: multiply the first terms. The first term in each bracket is multiplied first.
    2x × x = 2x2
  2. Outside: multiply the outer terms. The first term of the first bracket times the second term of the second.
    2x × (−3) = −6x
  3. Inside: multiply the inner terms. The second term of the first bracket times the first term of the second.
    1 × x = x
  4. Last: multiply the last terms. The two numbers at the end are multiplied.
    1 × (−3) = −3
  5. Collect the like terms. Add the two x terms together and leave the x squared and the number as they are.
    2x2 − 5x − 3

Check. Test with x = 1 in the brackets and in the answer: the two give the same value. ✓

Watch out. The two x terms are like terms, so combine them. Do not write x2 as 2x.

Watch out

Watch out: (x + 3)2 is not x2 + 9. It means (x + 3)(x + 3) = x2 + 6x + 9.

Exam tip

Grown-up tip. Ask the pupil to write the four products on separate lines before collecting. Most mistakes come from missing one of them.

Exercise 5C  Expanding two brackets

Non-calculator
1
FExpand (x + 3)(x + 4).
2
FExpand (x + 1)(x + 6).
3
FExpand (x + 2)(x − 5).
4
FExpand (x − 4)(x − 2).
5
FExpand (2x + 1)(x + 3).
6
FExpand (3x + 2)(x − 1).
7
FExpand (x + 5)(x − 5).
8
FExpand (2x − 3)(2x + 3).
9
FExpand (x + 4)2, which means (x + 4)(x + 4).
10
FExpand: (a) (x + 7)(x + 2) (b) (3x − 1)(2x + 5)
(a)(a) (x + 7)(x + 2)
(b)(b) (3x − 1)(2x + 5)
11
PSA rectangle has sides (x + 3) cm and (x + 5) cm. Write its area as an expression in x.
12
RIs (x + 2)(x + 3) equal to x2 + 6? Explain.
5.4

Factorising by taking out common factors

Year 7
A4
Video lesson: Factorising by taking out common factorsComing soon
A rectangle has area 6x + 9 square centimetres and one side of 3 cm. To find the other side, we need to write 6x + 9 as 3 times something. This is factorising: writing an expression as a product.

Factorising is the reverse of expanding. Find the biggest common factor: the biggest number and the letters that divide into every term. Write it outside a bracket, and divide each term by it to find what goes inside.

For example, 6x + 9 = 3(2x + 3), because 3 divides 6 and 9. Expanding 3(2x + 3) gives 6x + 9 again, so we can check.

Key ideas

  • Common factor: the largest number and letters that divide into every term.
  • Divide each term by the common factor to find the terms inside the bracket.
  • Check by expanding the answer: you should get the original expression back.
  • Factorising is the reverse of expanding.
Worked example 5.4.1
Factorise 6x + 9.

Method. Find the biggest number and the biggest letters that divide into every term. Write them outside the bracket.

  1. Find the common factor. Look at the numbers and at the letters in every term.
    common factor: 3
  2. Divide each term by the common factor. This gives the terms inside the bracket.
    6x + 9 ÷ 3 = 2x + 3
  3. Write the answer. The factor goes outside the bracket: multiply it back to check.
    3(2x + 3)

Check. 3 × (2x + 3) gives back 6x + 9. ✓

Watch out. Take out the biggest common factor, not just any common factor. Check by multiplying back.

Worked example 5.4.2
Factorise 4a2 − 6a.

Method. Find the biggest number and the biggest letters that divide into every term. Write them outside the bracket.

  1. Find the common factor. Look at the numbers and at the letters in every term.
    common factor: 2a
  2. Divide each term by the common factor. This gives the terms inside the bracket.
    4a2 − 6a ÷ 2a = 2a − 3
  3. Write the answer. The factor goes outside the bracket: multiply it back to check.
    2a(2a − 3)

Check. 2a × (2a − 3) gives back 4a2 − 6a. ✓

Watch out. Take out the biggest common factor, not just any common factor. Check by multiplying back.

Worked example 5.4.3
Factorise 12xy + 8x.

Method. Find the biggest number and the biggest letters that divide into every term. Write them outside the bracket.

  1. Find the common factor. Look at the numbers and at the letters in every term.
    common factor: 4x
  2. Divide each term by the common factor. This gives the terms inside the bracket.
    12xy + 8x ÷ 4x = 3y + 2
  3. Write the answer. The factor goes outside the bracket: multiply it back to check.
    4x(3y + 2)

Check. 4x × (3y + 2) gives back 12xy + 8x. ✓

Watch out. Take out the biggest common factor, not just any common factor. Check by multiplying back.

Watch out

Watch out: 6x + 9 = 3(2x + 3), not 3(2x + 9) or 3(2x) + 3. Every term must be divided by the factor, including the 9.

Exam tip

Grown-up tip. Ask the pupil to multiply the factor back into the bracket. If they get the original expression, the factorisation is right.

Exercise 5D  Factorising

Non-calculator
1
FFactorise 2x + 6.
2
FFactorise 5x + 10.
3
FFactorise 3x − 12.
4
FFactorise 6x + 9.
5
FFactorise 8a − 4.
6
FFactorise x2 + 3x.
7
FFactorise 4a2 − 6a.
8
FFactorise 7x − 14x2.
9
FFactorise 12xy + 8x.
10
FFactorise: (a) 10p + 15 (b) 9m2 − 3m (c) 5ab − 10a
(a)(a) 10p + 15
(b)(b) 9m2 − 3m
(c)(c) 5ab − 10a
11
PSThe area of a rectangle is 6x + 9 cm2. One side is 3 cm. Find the other side, in terms of x.
12
RSam factorises 4x + 8 as 4(x + 8). Is he right? Explain.
5.5

Multiplying and dividing terms

Year 7
A4
Video lesson: Multiplying and dividing termsComing soon
A rectangle has sides 2a cm and 5b cm. Its area is 2a × 5b. Multiplying terms uses the same rules as multiplying numbers, with the letters and their powers multiplied as well.

To multiply terms, multiply the numbers together and multiply the letters: 2a × 5b = 10ab, and 3x × 4x = 12x2. The powers add when the same letter is multiplied: x × x2 = x3.

To divide terms, divide the numbers and cancel the letters: 8p2 ÷ 2p = 4p.

Key ideas

  • Multiply the numbers, then multiply the letters. Powers of the same letter add: x × x2 = x3.
  • Divide the numbers, and cancel the same letters from top and bottom.
  • x × x = x2, not 2x.
  • Put the number first in the answer.
Worked example 5.5.1
Work out 2a × 5b.

Method. Group the like terms together, then add or subtract their numbers. Keep the letter and the power the same.

  1. Multiply the numbers. 2 × 5 = 10.
    10
  2. Multiply the letters. a × b is written ab.
    ab
  3. Write the answer. Put the number first.
    10ab

Check. With a = 1 and b = 1: 2 × 5 = 10 and 10 × 1 = 10. ✓

Watch out. Do not write 10a + b.

Worked example 5.5.2
Work out 3x2 × 4x.

Method. Group the like terms together, then add or subtract their numbers. Keep the letter and the power the same.

  1. Multiply the numbers. 3 × 4 = 12.
    12
  2. Multiply the x terms. x2 × x = x3, because the powers add: 2 + 1 = 3.
    x3
  3. Write the answer. Put the number first.
    12x3

Check. With x = 2: 3x2 = 12 and 4x = 8, so 12 × 8 = 96, and 12x3 = 12 × 8 = 96. ✓

Watch out. The powers add, but the numbers multiply.

Worked example 5.5.3
Work out 8p2 ÷ 2p.

Method. Group the like terms together, then add or subtract their numbers. Keep the letter and the power the same.

  1. Divide the numbers. 8 ÷ 2 = 4.
    4
  2. Cancel one p from top and bottom. p2 ÷ p = p, because one p cancels.
    p2 ÷ p = p
  3. Write the answer. The answer is 4p.
    4p

Check. With p = 3: 8 × 9 ÷ 6 = 12, and 4 × 3 = 12. ✓

Watch out. Dividing by p removes one p from p2, leaving p.

Watch out

Watch out: x × x is x2, not 2x. Adding and multiplying are different: x + x = 2x, but x × x = x2.

Exam tip

Grown-up tip. Ask the pupil to test the answer with x = 2. The working is then a number check, which is easier to trust.

Exercise 5E  Multiplying and dividing terms

Non-calculator
1
FWork out 3a × 4.
2
FWork out 2x × 5y.
3
FWork out 4m × 3m.
4
FWork out x × x2.
5
FWork out 6y2 × 2y.
6
FWork out 8p2 ÷ 2p.
7
FWork out 12k ÷ 3.
8
FWork out 15a2b ÷ 5a.
9
FWork out 2x × 3x × 4y.
10
FWork out: (a) 2x × 3x (b) 10y ÷ 5 (c) a2b × ab
(a)(a) 2x × 3x
(b)(b) 10y ÷ 5
(c)(c) a2b × ab
11
PSA rectangle has sides 3x cm and 4x cm. Write its area in terms of x.
12
RSam says 2x × 3x = 6x. Is he right? Explain.

Chapter summary

SkillMethod
Like termsSame letter and same power; add or subtract their numbers and keep the letter
Expanding a bracketMultiply the term outside by every term inside; keep the signs
Expanding two bracketsFOIL: First, Outside, Inside, Last; then collect like terms
FactorisingTake out the highest common factor; divide each term by it; check by expanding
Multiplying termsMultiply the numbers, multiply the letters; powers of the same letter add
Dividing termsDivide the numbers; cancel the same letters from top and bottom

Can you…?

  • I can collect like terms to simplify an expression
  • I can expand a single bracket, including with a negative number or a letter outside
  • I can expand two brackets using FOIL, and collect the terms
  • I can factorise an expression by taking out the highest common factor
  • I can multiply and divide terms with numbers and powers of letters

Key terms

like terms
terms with the same letter and the same power, such as 3x and 5x
expand
multiply out a bracket so that there are no brackets left
factorise
write an expression as a product, with a common factor outside a bracket
common factor
a number or letter that divides into every term
FOIL
a method for expanding two brackets: First, Outside, Inside, Last
equivalent
two expressions that give the same answer for every value of the letter

Exercise 5Z  Check-up

Non-calculator
1
Simplify 4a + 3a − a.
2
Expand 2(x − 4).
3
Expand x(3x + 2).
4
Expand (x + 5)(x + 3).
5
Expand (x + 6)(x − 2).
6
Factorise 9x − 15.
7
Factorise 4x2 + 8x.
8
Work out 2a × 7b.
9
Work out 20p ÷ 4.
10
Expand −3(2x − 5).

Total 19 marks. Allow about 19 minutes.

Exercise 5X  Practice test questions

Non-calculator
1
FSimplify 5a + 2a − 3a.[1 mark]
2
FExpand 4(x + 3).[1 mark]
3
FFactorise 6x + 3.[1 mark]
4
FExpand and simplify 2(x + 5) + 3(x − 1).[2 marks]
5
FExpand (x + 4)(x + 2).[2 marks]
6
FFactorise fully 10a2 + 15a.[2 marks]
7
FSimplify 3x × 5x.[2 marks]
8
FSimplify 12p2 ÷ 4p.[2 marks]
9
FExpand and simplify (2x − 1)(x + 4).[3 marks]
10
RShow that 3(x + 2) + 2(x + 1) is the same as 5x + 8. Explain each step.[3 marks]
← Chapter 4: Algebraic notation and substitutionChapter 6: Solving equations and using formulae →

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