Year 8 · Unit 4 Expressions, equations and inequalities · Lesson 2 of 6

Can you run expanding backwards?

Factorizing writes a sum as a product: take out the highest common factor, put what is left in a bracket, and check by expanding.

Number and AlgebraNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Lesson 1 of this unit. Expand 5(2x − 3).
    Answer10x − 15. 5 × 2x = 10x and 5 × (−3) = −15.
  2. From Year 7, Unit 1, lesson 5, common factors. What is the highest common factor of 24 and 36?
    Answer12. It is the largest number that divides both.
  3. From Unit 3 (integers), lesson 4. Work out (−20) ÷ (−5).
    Answer4. Two negative numbers divide to a positive.
Words used in this lesson
factor
a number that divides another exactly. For example: 6 is a factor of 18.
common factor
a factor of two or more numbers. For example: 3 is a common factor of 12 and 18.
highest common factor
the largest of the common factors. For example: 6 for 12 and 18.
factorize
write a sum as a product with a bracket. For example: 12x + 18 = 6(2x + 3).
expand
multiply out a bracket. For example: 6(2x + 3) = 12x + 18.

Running expanding backwards

Last lesson, 4(x + 3) became 4x + 12. Now run it the other way. A teacher in Lagos has 12 boxes with x pencils in each box, and 18 loose pencils: 12x + 18 pencils in all. She wants to share them into identical groups with none left over, without opening any box, and she wants as many groups as possible. How many groups can she make?

Predict first: what is the largest number that divides both 12 and 18? Then see how the idea works on other pairs.

20
30
10highest common factor of p and q
2x + 3what is left in the bracket

20x + 30: the highest common factor of 20 and 30 is 10. 20x is 2 groups of 10x and 30 is 3 groups of 10, so 20x + 30 = 10(2x + 3).

Show me the steps
  1. p = 20, q = 30: the common factors are 1, 2, 5 and 10; the highest is 10, so 20x + 30 = 10(2x + 3).
  2. p = 18, q = 24: the highest is 6, so 18x + 24 = 6(3x + 4).
  3. p = 6, q = 6: the highest is 6, and 6x is one group of 6x, so 6x + 6 = 6(x + 1). A 1 stays in the bracket.
  4. p = 7, q = 9: the only common factor is 1, so 7x + 9 cannot be factorized.

Factorizing is expanding backwards. Find the highest common factor of the terms, write it outside a bracket, and divide each term by it to fill the bracket.

Check by expanding: 10(2x + 3) = 20x + 30. If you do not get the original back, something has gone wrong.

Which number to take out?

You can take out any common factor, but only the highest one finishes the job. Use the slider on 12x + 18.

3
0left over from 12x and 18
not finishedfactorizing with f = 3

f = 3: 12x + 18 = 3(4x + 6). That is true, but 4 and 6 still share a factor of 2, so it is not finished.

Show me the steps
  1. f = 3: 12x + 18 = 3(4x + 6). True, but 4 and 6 share a factor of 2, so it is not finished.
  2. f = 6: 12x + 18 = 6(2x + 3). The 2 and 3 share only 1, so this is fully factorized.
  3. f = 4: 12 is a multiple of 4 but 18 is not (18 ÷ 4 leaves 2), so 4 is not a common factor.
  4. f = 1: 12x + 18 = 1(12x + 18). True, but nothing has been taken out.

Back to the pencils: the most groups is 6, each with 2 boxes and 3 loose pencils, so 12x + 18 = 6(2x + 3). Three groups would also work, 3(4x + 6), but each group could still be split in two.

Two slips to watch for. 12x + 18 = 3(4x + 6) is true, but it is not finished because 4 and 6 still share a factor of 2. And when you take out a whole term, a 1 is left behind: 4x + 4 = 4(x + 1), not 4(x).

Worked examples

Factorize 15x + 25.

  1. The highest common factor of 15 and 25 is 5.
  2. 15x ÷ 5 = 3x and 25 ÷ 5 = 5, so 5(3x + 5).
  3. Check: 5 × 3x + 5 × 5 = 15x + 25.

Factorize 14x − 21. The highest common factor of 14 and 21 is 7. 14x ÷ 7 = 2x and −21 ÷ 7 = −3, so 7(2x − 3).

Factorize 6x + 6. The highest common factor is 6. 6x ÷ 6 = x and 6 ÷ 6 = 1, so 6(x + 1). The 1 matters: 6(x) is 6x, not 6x + 6.

Your go. The first step is given. Factorize 18a + 24 fully.

  1. The highest common factor of 18 and 24 is 6.
  2. Divide each term by 6 to fill the bracket, then check by expanding.
Check the answer

6(3a + 4). Check: 6 × 3a = 18a and 6 × 4 = 24.

Check before you go on

1. Which is 9x + 12 factorized fully?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. 8x + 20 = 4(2x + k). What is k?

3. What is the highest common factor of 18 and 45?

Level 2 · Building

4. 16x − 24 = 8(2x − k). What is k?

5. Which of these equals 18x + 30 but is not fully factorized?

6. 7x + 7 = 7(x + k). What is k?

Level 3 · Stretch

7. A rectangular floor has area (15x + 35) square metres and is 5 m long. Its width is (3x + k) metres. What is k?

8. −2x − 6 = −2(x + k). What is k?

Something to think about

Not every expression of the form px + q can be factorized by taking out a number. Use the first figure to find three different pairs of p and q that cannot. What do they have in common?

One way to think about it

Any pair with a highest common factor of 1 works, for example 7x + 9, 8x + 15 and 10x + 21. The only number that divides both is 1, so there is no useful common factor.

Exit ticket

9. Which is 28x − 42 factorized fully?