Common Factors (SQA National 5 Maths): Revision Note

Factorising expressions with a common factor

What is factorisation?

• A factorised expression is one written as the product (multiplication) of two or more terms (factors)

  • $3(x+2)$ is factorised

    • It is $3\times (x+2)$

  • $3x+6$  is not factorised

  • $3xy$  is factorised

    • It is $3\times x\times y$

  • Numbers can also be factorised

    • $12=2\times 2\times 3$

• In algebra, factorisation is the reverse of expanding brackets

  • It's putting it into brackets, rather than removing brackets

How do I factorise two terms?

• To factorise $12x^2+18x$

  • Find the highest common factor of the number parts

    • 6

  • Find the highest common factor of the algebra parts

    • $x$

  • Multiply both to get the overall highest common factor

    • $6x$

  • $12x^2+18x$  is the same as $6x\times 2x+6x\times 3$

    • Using the highest common factor

  • Take out the highest common factor

    • Write it outside a set of brackets

    • Put the remaining terms, $2x+3$,  inside the brackets

  • This gives the answer

    • $6x(2x+3)$

• To factorise an expression containing multiple variables, e.g. $2a^{3}b-4a^2{b}^2$

  • Use the same approach as above

  • Find the highest common factor of the number parts

    • 2

  • Find the highest common factor of the algebra parts

    • $a$ and $b$ appear in both terms

      • The highest common factor of $a^3$ and $a^2$ is $a^2$

      • The highest common factor of $b$ and $b^2$ is $b$

    • $a^{2}b$

  • Multiply both to get the overall highest common factor

    • $2a^{2}b$

  • $2a^{3}b-4a^2{b}^2$  is the same as $2a^{2}b\times a-2a^{2}b\times 2b$

    • Using the highest common factor

  • Take out the highest common factor

    • Write it outside a set of brackets

    • Put the remaining terms, $a-2b$,  inside the brackets

  • This gives the answer

    • $2a^{2}b(a-2b)$