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Factorising

Factorising

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Explainer Video

Tutor: Toby

Summary

Factorising

​​​​In a nutshell

Factorising is the inverse process of expanding brackets. Factorising helps you to find the roots of equations and they offer an alternative, sometimes more useful, way to write expressions. 


Factorising a linear expression

To factorise a linear expression, find all common factors between each term and factor them out. As a result, divide each term by these factors. Write the result inside brackets, with the factored out factors sitting before the brackets.


Example 1

Factorise fully 4x+324x+324x+32.


There is a common factor of 444 between these terms. Hence:

​4x+32=4(x+8)‾4x+32=\underline{4(x+8)}4x+32=4(x+8)​​​


This method can work with non-linear expressions.


Example 2

Factorise fully 5x3y2−10xy3+20xy25x^3y^2-10xy^3+20xy^25x3y2−10xy3+20xy2.


Each term has a common factor of 5xy25xy^25xy2. Hence the factorised form is:

​5x3y2−10xy3+20xy2=5xy2(x2−2y+4)‾5x^3y^2-10xy^3+20xy^2=\underline{5xy^2(x^2-2y+4)}5x3y2−10xy3+20xy2=5xy2(x2−2y+4)​​​


This is fully factorised since there are no more common factors between the terms.



Factorising a quadratic expression

PROCEDURE

​1.

Ensure the quadratic is in the form ax2+bx+cax^2 + bx + cax2+bx+c and identify a,ba,ba,b and ccc​​

2.

Set up the answer with two brackets, as follows (Ax+B)(Cx+D)(Ax +B)(Cx +D)(Ax+B)(Cx+D)​​

3.

Find two numbers AAA​ and CCC which multiply to give aaa, and also two numbers BBB and DDD that multiply to give ccc such that when the brackets are expanded, ACACAC​ and BDBDBD add to give bbb​​

4.

Fill these numbers in the double brackets 


Example 3

Factorise fully x2+12x+20x^2+12x+20x2+12x+20.


You have that a=1a=1a=1, b=12b=12b=12 and c=20c=20c=20. Set up the factorisation:

x2+12x+20=(Ax+B)(Cx+D)x^2+12x+20=(Ax+B)(Cx+D)x2+12x+20=(Ax+B)(Cx+D)​​


Since a=1a=1a=1, you have that AC=1AC=1AC=1. Set A=C=1A=C=1A=C=1. 
x2+12x+20=(x+B)(x+D)x^2+12x+20=(x+B)(x+D)x2+12x+20=(x+B)(x+D)​​


Now seek a BBB and DDD such that B+D=12B+D=12B+D=12 and BD=20BD=20BD=20. You can use B=2B=2B=2 and D=10D=10D=10:

x2+12x+20=(x+2)(x+10)‾x^2+12x+20=\underline{(x+2)(x+10)}x2+12x+20=(x+2)(x+10)​​​


This is the fully factorised form.


There are some notable product that can help with factorisation:

  • ​(a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2(a+b)2=a2+2ab+b2
  • ​(a−b)2=a2−2ab+b2(a-b)^2 = a^2 -2ab + b^2(a−b)2=a2−2ab+b2​​
  • ​(a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2(a+b)(a−b)=a2−b2

Example 4

Factorise this expression as far as possible: 4x2−9y24x^2 - 9y^24x2−9y2.


This is similar to one of the notable products: a2−b2=(a+b)(a−b)a^2 - b^2 = (a+b)(a-b)a2−b2=(a+b)(a−b). Equate a2=4x2a ^2= 4x^2a2=4x2 and b2=9y2b^2 = 9y^2b2=9y2, therefore a=2xa = 2xa=2x and b=3yb = 3yb=3y. 


Now write your factorised expression:

​4x2−9y2=(2x+3y)(2x−3y)‾4x^2-9y^2 =\underline{(2x + 3y)(2x-3y)}4x2−9y2=(2x+3y)(2x−3y)​​​


Factorising simple cubic expressions

Some cubic expressions can be factorised partially, leaving an expression you already know how to factorise.


Example 5

Factorise this expression as far as possible: x3+3x2−10xx^3 + 3x^2 - 10xx3+3x2−10x.


First, extract the common factor xxx.

x3+3x2−10x=x(x2+3x−10)x^3 + 3x^2 -10x = x(x^2 + 3x -10)x3+3x2−10x=x(x2+3x−10)​​


Now you have a quadratic expression. Factorising this can be done by looking for two numbers that add to three and multiply to minus ten. The numbers minus five and two do this:

Therefore x3+3x2−10=x(x−5)(x+2)‾x^3 + 3x^2 -10 = \underline{x(x-5)(x+2)}x3+3x2−10=x(x−5)(x+2)​​


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Expanding double brackets: FOIL and multiplication grid

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Expanding double brackets: FOIL and multiplication grid

Double brackets: Expanding and factorising

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Double brackets: Expanding and factorising

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Factorising

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Factorising

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FAQs - Frequently Asked Questions

What is factorising?

Factorising is, essentially, the inverse process of expanding brackets.

How do I know when to stop factorising?

You should stop factorising when everything in your expression has the form (x-root) or (x+root), or when the terms that are left in your expression cannot be simplified.

What is the first rule of factorising?

Find the common factor and extract it before doing anything else. This could simplify the process.

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