Brackets and Factorising · Maths Practice

Key ideas

Expanding

Multiply every term inside the bracket by the term outside.

Use a grid: the outside term down the side, the inside terms along the top.

+12

3(x + 4) = 3x + 12

Collecting like terms

Like terms have exactly the same letters: 5x and −2x are like terms; 5x and 5 are not.

Add or take away the numbers in front: 5x − 2x = 3x.

Factorising

Factorising is expanding backwards.

1. Find the highest common factor (HCF).

2. Write it outside a bracket.

3. Divide each term by it to fill the bracket.

6x + 9 = 3(2x + 3)

Check

Expand your factorised answer. You should get back to the question.

If the terms in the bracket still have a common factor, you haven't finished.

Typing answers: type 3x + 12 or 3(2x + 3). Use the ² button (or ^2) for squares, like x² + 5x. Check the preview under the box.

Expanding a single bracket

Brackets

Expanding a single bracket

Expand 4(2x − 5).(2 marks)

Draw the grid.

−5

In plain EnglishWrite the 4 down the side and the two terms in the bracket, 2x and −5, along the top. Keep the sign with each term.

Multiply to fill each cell.

−5

−20

= 8x − 20✓¹Both terms correct✓²

In plain English4 × 2x = 8x and 4 × −5 = −20. Then write the terms out in a line.

⚠ A common mistake: only multiplying the first term: 8x − 5. Every term in the bracket gets multiplied.

Marking points

M1 8x

M2 −20

Brackets

A letter outside the bracket

Expand a(a + 6).(2 marks)

Grid.

a²

+6a

a × a = a²✓¹

In plain Englisha × a = a² (a squared). a × 6 = 6a.

Write it out.= a² + 6a✓²

In plain EnglishWrite the two answers from the grid with a + between them. a² and 6a are not like terms, so this is as simple as it gets.

Marking points

M1 a²

M2 +6a

Expanding two brackets and collecting like terms

Brackets

Two brackets, then collect like terms

Expand and simplify 3(x + 4) + 2(x − 1).(3 marks)

Expand the first bracket.

+12

3x + 12✓¹

In plain English3 × x = 3x and 3 × 4 = 12.

Expand the second bracket.

−1

−2

3x + 12 + 2x − 2✓²

In plain English2 × x = 2x and 2 × −1 = −2. Write all four terms in a line.

Collect like terms.3x + 2x = 5x x terms12 − 2 = 10 numbers= 5x + 10✓³

In plain EnglishPut the x terms together and the numbers together.

Marking points

M1 3x + 12

M2 2x − 2

M3 5x + 10

Brackets

Taking a bracket away

Expand and simplify 5(2y + 3) − 2(y + 4).(3 marks)

First bracket.

10y

+15

10y + 15✓¹

In plain EnglishMultiply each term in the first bracket by 5: 5 × 2y = 10y and 5 × 3 = 15.

Second bracket: the − goes with the 2.

−2

−2y

−8

10y + 15 − 2y − 8✓²

In plain EnglishThe number outside is −2, so −2 × y = −2y and −2 × 4 = −8.

Collect like terms.10y − 2y = 8y, 15 − 8 = 7= 8y + 7✓³

In plain EnglishPut the y terms together and the numbers together. Be careful with the signs: it is − 2y and − 8.

⚠ A common mistake: writing −2y + 8. The minus belongs to the 2, so it multiplies the 4 as well.

Marking points

M1 10y + 15

M2 −2y − 8

M3 8y + 7

Factorising: a number outside

Brackets

Factorising with a number

Factorise 6x + 9.(2 marks)

Find the HCF.Factors of 6: 1, 2, 3, 6Factors of 9: 1, 3, 9HCF = 3✓¹

In plain EnglishThe highest number that goes into both 6 and 9 is 3.

Divide each term by the HCF.6x3 = 2x, 93 = 36x + 9 = 3(2x + 3)✓²Check: 3 × 2x = 6x, 3 × 3 = 9 ✓

In plain EnglishPut the 3 outside the bracket. Inside goes what is left of each term after dividing by 3.

Marking points

M1 HCF = 3

M2 3(2x + 3)

Brackets

Make sure it is fully factorised

Factorise 12a − 18.(2 marks)

Find the highest common factor.2, 3 and 6 all go into 12 and 18HCF = 6✓¹

In plain EnglishIf you used 2 or 3, there would still be a common factor left inside the bracket.

Divide each term.12a6 = 2a, 186 = 36(2a − 3)✓²

In plain EnglishDivide each term by 6 to find what goes inside the bracket. Check by expanding: 6 × 2a = 12a and 6 × 3 = 18.

⚠ Not fully factorised: 2(6a − 9) is equal, but 6 and 9 still share a factor of 3. Always use the highest common factor.

Marking points

M1 HCF = 6

M2 6(2a − 3)

Factorising: a letter outside

Brackets

A letter as the common factor

Factorise x² + 7x.(2 marks)

Find the common factor.x² = x × x, 7x = 7 × xHCF = x✓¹

In plain EnglishBoth terms have an x in them.

Divide each term by x.x²x = x, 7xx = 7x(x + 7)✓²

In plain EnglishPut x outside and what is left inside. Check by expanding: x × x = x² and x × 7 = 7x.

Marking points

M1 HCF = x

M2 x(x + 7)

Brackets

A number and a letter

Factorise 8y² + 12y.(2 marks)

Numbers and letters.HCF of 8 and 12 is 4Both terms have a yHCF = 4y✓¹

In plain EnglishTake out the biggest number and the letter that both terms share.

Divide each term by 4y.8y²4y = 2y, 12y4y = 34y(2y + 3)✓²

In plain EnglishPut 4y outside and what is left inside. Check by expanding: 4y × 2y = 8y² and 4y × 3 = 12y.

Marking points

M1 HCF = 4y

M2 4y(2y + 3)

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Your turn

Type your answers like 3x + 12 or 3(2x + 3), and check the preview under the box. Factorising answers must be fully factorised. Stuck? Tap Need a hint?