What you need to know

Things to remember:

• Factors of a number are numbers we can multiply together to make the original.
• We can factorise by looking for a common factor and then putting this outside a bracket.
• Choosing the highest common factor will factorise fully.

This topic is all to do with factors, but this time we will be looking at algebraic factors. But, before we start, let’s just have a look back at what factors are.

We need to remember that factors can be thought of in two ways:

• Factors are two whole numbers we multiply to make another number.

$$4\times8=32$$

4 and 8 are factors of 32

• We can also do the opposite, division. If we get a whole number when dividing, then these are factors.

$$32\div2=16$$

2 and 16 are factors of 32

It is also important that we remember how to expand a single bracket, i.e. we multiply everything inside the bracket by what is outside.

$$5(2x+3)=5\times2x+5\times3=10x+15$$

$$2(3-4y)=2\times3-2\times4y=6-8y$$

$$3a(a+b)=3a\times a + 3a\times b= 3a^2+3ab$$

Now, let’s get to the heart of this topic, factorising.

Factorise fully $9x+15$.

Factorising is the opposite of expanding a bracket, i.e. we are putting it back into brackets, and we do it in 4 steps.

Step 1: Find the factors of the terms.

Factors of 15

1                      15

3                      5

Hint: To find the factors of an algebraic term, list all of the factors of the number and then multiply by the variable.

Factors of 9x

1                      9x

3                      3x

9                      a

Step 2: Find the highest common factor from both sets of factors.

The highest common factor is 3.

Step 3: Look at the highest common factors factor pairs.

Factors of 15

3                      5

Factors of 9a

3                      3x

Step 4: Put into factorise form.

The number from Step 2 goes outside the bracket.

The numbers from Step 3 go inside the bracket.

$$9x+15=3(3x+5)$$

We just need to be careful if there is a minus.

Factorise fully $7x-42$.

Step 1: Find the factors of the terms.

Factors of 42

1                      42

2                      21

3                      14

6                      7

Hint: To find the factors of an algebraic term, list all of the factors of the number and then multiply by the variable.

Factors of 7x

1                      7x

7                      x

Step 2: Find the highest common factor from both sets of factors.

The highest common factor is 7.

Step 3: Look at the highest common factors factor pairs.

Factors of 42

6                      7

Factors of 7x

7                      x

Step 4: Put into factorise form.

The number from Step 2 goes outside the bracket.

The numbers from Step 3 go inside the bracket.

Remember, we want to have -42, so need to use -6.

$$7x-42=7(x-6)$$

Example Questions

Question 1: Factorise fully $12x+8$.

Step 1: Find the factors of the terms.

Factors of 8

1                      8

2                      4

Hint: To find the factors of an algebraic term, list all of the factors of the number and then multiply by the variable.

Factors of 12x

1                      12x

2                      6x

3                      4x

4                      3x

6                      2x

12                    x

Step 2: Find the highest common factor from both sets of factors.

The highest common factor is 4.

Step 3: Look at the highest common factors factor pairs.

Factors of 8

2                      4

Factors of 7x

4                      3x

Step 4: Put into factorise form.

The number from Step 2 goes outside the bracket.

The numbers from Step 3 go inside the bracket.

$$12x+8=4(3x+2)$$

Question 2: Factorise fully $21x-14$.

Step 1: Find the factors of the terms.

Factors of 14

1                      14

2                      7

Hint: To find the factors of an algebraic term, list all of the factors of the number and then multiply by the variable.

Factors of 21x

1                      21x

3                      7x

7                      3x

21                    x

Step 2: Find the highest common factor from both sets of factors.

The highest common factor is 7.

Step 3: Look at the highest common factors factor pairs.

Factors of 14

2                      7

Factors of 7x

7                      3x

Step 4: Put into factorise form.

The number from Step 2 goes outside the bracket.

The numbers from Step 3 go inside the bracket.

Remember, we want to have -14, so need to use -2.

$$21x-14=7(3x-2)$$