Factorise 14x^2 − 12x. What do you get? A. 2(7x^2 − 6x) B. 2x(7x − 6) C. 2x(7x − 12) D. x(14x − 12) Choose the right answer

B. 2x(7x − 6). The H.C.F. of 14 and 12 is 2, and both terms have x: the common factor is 2x. 14x^2 ÷ 2x = 7x and 12x ÷ 2x = 6.

x(y + 8) + 8(y + 8) = (y + □)(x + □) Fill in each blank

8, 8. Both terms have the bracket (y + 8): take the whole of it out, (y + 8)(x + 8).

56x^2 + 7x = 7x(□x + □) Fill in each blank

8, 1. 56x^2 ÷ 7x = 8x and 7x ÷ 7x = 1: taking out a whole term leaves 1 behind.

Factorisation · Ages 13–14

Taking out a common factor

The coefficients' H.C.F., each letter to its lowest index.

How to do it

Method

The coefficients' H.C.F., each letter to its lowest index

6x² − 9x = 3x(2x − 3)

Another way

Expand to check

3x(2x − 3) = 6x² − 9x

In full

Factorising writes a polynomial as factors multiplied: the reverse of expanding. Find what every term shares first: the H.C.F. of the coefficients, and each letter every term has, to its lowest index. 6x² − 9x: the H.C.F. of 6 and 9 is 3, every term has x, at least x¹, so the common factor is 3x and 6x² − 9x = 3x(2x − 3). Each term inside the bracket is the old term divided by the common factor: 6x² ÷ 3x = 2x, 9x ÷ 3x = 3. A common factor can be a whole bracket: x(y + 2) + 3(y + 2) = (y + 2)(x + 3). Taking out a whole term leaves 1 behind: 4x + 2 = 2(2x + 1). Expand to check you are back where you began, and look inside the bracket for anything more: 2x(4x − 6) still has a 2 to take out.

Worked examples

Have a go first, then open each one for the answer and the working.

Example 1
Factorise 14x² − 12x. What do you get?
- A. 2(7x² − 6x)
- B. 2x(7x − 6)
- C. 2x(7x − 12)
- D. x(14x − 12)
Choose the right answer
Show the answer and working
Answer: B. 2x(7x − 6)
The H.C.F. of 14 and 12 is 2, and both terms have x: the common factor is 2x. 14x² ÷ 2x = 7x and 12x ÷ 2x = 6.
Example 2
x(y + 8) + 8(y + 8) = (y + )(x + )
Fill in each blank
Show the answer and working
Answer: 8, 8
Both terms have the bracket (y + 8): take the whole of it out, (y + 8)(x + 8).
Example 3
56x² + 7x = 7x(x + )
Fill in each blank
Show the answer and working
Answer: 8, 1
56x² ÷ 7x = 8x and 7x ÷ 7x = 1: taking out a whole term leaves 1 behind.

Practice questions to try

Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.

1.49x + 35 = 7(x + )
Fill in each blank
Answer:
2.Factorise 45x² − 25x. What do you get?
- A. 5x(9x − 25)
- B. 5(9x² − 5x)
- C. 5x(9x − 5)
- D. x(45x − 25)
Choose the right answer
Answer:
3.35x²y − 30xy² = 5xy(7x − 6y)
True or false?
Answer:
4.10x³ − 2x² = x²(x − )
Fill in each blank
Answer:
5.Factorise 8x(x − 3) − 9(x − 3). What do you get?
- A. (8x − 9)(x + 3)
- B. (x − 3)(8x + 9)
- C. (x − 3)(8x − 9)
- D. (x − 3)(8x − 9x)
Choose the right answer
Answer:

All the answers

1. 7, 5
2. C. 5x(9x − 5)
3. True
4. 2, 5, 1
5. C. (x − 3)(8x − 9)

Free worksheet: Taking out a common factorWorking out, multiple choice and word problems, with answers and working, to print or save as a PDF

VIP practice notes and papers

Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.

Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
Practice paper: Set out in parts like a school paper, marked, with answers and full working

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