Algebra
Grade 4-5Factorising
Factorising is the reverse of expanding brackets: instead of multiplying a bracket out, the aim is to find the highest common factor of every term and write it outside a bracket. This lesson covers finding a common numerical factor, factorising with a letter factor, factorising with a negative common factor, and factorising expressions with two variables.
Written by Asad, Co-Founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Finding a common numerical factor
Find the highest number that divides into every term, write it outside a bracket, and divide each term by it to find what goes inside.
6x + 9 has a highest common factor of 3, since 3 divides into both 6 and 9. Dividing each term by 3 gives 2x and 3, so the factorised form is 3(2x + 3).
Factorising with a letter factor
When every term shares a letter, that letter can be factored out too, alongside any common number.
x² + 5x has a common factor of x, since x² = x × x and 5x = x × 5. Factoring out x leaves x + 5, giving x(x + 5).
Factorising with a negative common factor
If every term is negative, factor out a negative common factor, remembering that dividing a negative term by a negative number gives a positive term.
−4x − 12 has a highest common factor of −4. Dividing each term by −4 gives x and 3, so the factorised form is −4(x + 3).
Factorising expressions with two variables
Find the highest common factor across both the numbers and the letters, checking each letter separately to see if it appears in every term.
6xy + 9x has a common numerical factor of 3, and x appears in both terms, so the highest common factor is 3x. Dividing each term by 3x gives 2y and 3, so the factorised form is 3x(2y + 3).
Worked Examples
Three exam-style questions, fully solved.
Factorise fully 6x + 9.
Easy- 1.Find the highest common factor of 6 and 9: 3
- 2.Divide each term by 3: 6x ÷ 3 = 2x, and 9 ÷ 3 = 3
Answer: 3(2x + 3)
Factorise fully 6x² + 9x.
Medium- 1.Find the highest common numerical factor of 6 and 9: 3
- 2.Check for a common letter factor: x appears in both terms
- 3.Combine to find the highest common factor: 3x, then divide each term by it: 6x² ÷ 3x = 2x, and 9x ÷ 3x = 3
Answer: 3x(2x + 3)
Factorise fully 6xy + 9x.
Hard- 1.Find the highest common numerical factor of 6 and 9: 3
- 2.Check for a common letter factor: x appears in both terms, but y does not
- 3.Combine to find the highest common factor: 3x, then divide each term by it: 6xy ÷ 3x = 2y, and 9x ÷ 3x = 3
Answer: 3x(2y + 3)
Avoid These
The most common mistakes students make.
Not finding the highest common factor, and instead factoring out a smaller factor, which leaves the expression not fully factorised.
Getting the sign wrong when factoring out a negative number, forgetting that dividing a negative term by a negative factor gives a positive term inside the bracket.
Forgetting to include a letter that appears in every term as part of the common factor, or including a letter that does not actually appear in every term.
Not checking the answer by expanding the bracket back out to confirm it matches the original expression.
When two different letters are involved, only factoring out the common number and forgetting to check whether a letter is also common to every term.
FAQ
Questions parents and students ask.
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