xy + 5x + 8y + 40 = (x + □)(y + □) Fill in each blank

8, 5. In pairs: x(y + 5) + 8(y + 5) = (x + 8)(y + 5).

xy + x + 7y + 7 = x(y + □) + □(y + 1) Fill in each blank

1, 7. xy + x = x(y + 1) and 7y + 7 = 7(y + 1): the two brackets match.

xy + x + 5y + 5 can be factorised as (x + 1)(y + 5). True or false?

False. xy + x + 5y + 5 = x(y + 1) + 5(y + 1) = (x + 5)(y + 1).

Factorisation · Ages 13–14

Factorising by grouping

Pairs, a factor out of each, the brackets must match.

How to do it

Method

Pairs, a factor out of each, the brackets must match

xy + 3x + 2y + 6 = (y + 3)(x + 2)

Another way

Group the other way

(xy + 2y) + (3x + 6) = y(x + 2) + 3(x + 2)

In full

When four terms share no factor, group them in pairs, take out each pair's common factor, then take out the bracket they share. xy + 3x + 2y + 6 = (xy + 3x) + (2y + 6) = x(y + 3) + 2(y + 3) = (y + 3)(x + 2). Taking a negative out of the second pair, watch the signs: xy − 3x − 2y + 6 = x(y − 3) − 2(y − 3) = (y − 3)(x − 2). The two brackets must be exactly the same; if not, try another grouping. Or group the other way: (xy + 2y) + (3x + 6) = y(x + 2) + 3(x + 2) = (x + 2)(y + 3). The answer is the same.

Worked examples

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

Example 1
xy + 5x + 8y + 40 = (x + )(y + )
Fill in each blank
Show the answer and working
Answer: 8, 5
In pairs: x(y + 5) + 8(y + 5) = (x + 8)(y + 5).
Example 2
xy + x + 7y + 7 = x(y + ) + (y + 1)
Fill in each blank
Show the answer and working
Answer: 1, 7
xy + x = x(y + 1) and 7y + 7 = 7(y + 1): the two brackets match.
Example 3
xy + x + 5y + 5 can be factorised as (x + 1)(y + 5).
True or false?
Show the answer and working
Answer: False
xy + x + 5y + 5 = x(y + 1) + 5(y + 1) = (x + 5)(y + 1).

Practice questions to try

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

1.xy + 6x + 4y + 24 = (x + )(y + )
Fill in each blank
Answer:
2.xy + 3x + 6y + 18 can be factorised as (x + 3)(y + 6).
True or false?
Answer:
3.Factorise xy − 5x − 3y + 15. What do you get?
- A. (x + 3)(y − 5)
- B. (x − 3)(y + 5)
- C. (x − 3)(y − 5)
- D. (x − 5)(y − 3)
Choose the right answer
Answer:
4.Factorise x³ + 4x² + 8x + 32. What do you get?
- A. (x + 8)(x² + 4)
- B. (x + 4)(x² − 8)
- C. (x + 4)(x² + 8)
- D. (x − 4)(x² + 8)
Choose the right answer
Answer:
5.8x + 8y − 4x − 4y = (x + y)
Answer:

All the answers

1. 4, 6
2. False
3. C. (x − 3)(y − 5)
4. C. (x + 4)(x² + 8)
5. 4

Free worksheet: Factorising by groupingWorking 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

Start playingPlay and learn in the secondary world, Star Galaxy: animated methods, monster battles, a review of every mistake