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Stage 5.1-2

2.10 Factorising binomial products

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Introduction

We saw how to use the distributive law to expand binomial products. We can also use it to factorise binomial products. In order to find what the factors are, we factorise the terms in pairs.

Factorise special binomial expressions

We can factorise the product of two binomial expressions using the rule: \left(A+B\right) \left(C+D\right)=AC+AD+BC+BD.

There are two special cases of factorising these expressions: \left(A+B\right)^{2}=A^{2}+2AB+B^{2} called a perfect square, and \left(A+B\right) \left(A-B\right)=A^{2}-B^{2} called difference of two squares.

Examples

Example 1

Fully factorise: 5\left(a+b\right) + v\left(a+b\right)

Worked Solution
Create a strategy

Factorise using the law, AB+AC=A\left(B+C\right).

Apply the idea

Both terms have a common factor of (a+b) so we can factorise this out.

\displaystyle 5\left(a+b\right) + v\left(a+b\right)\displaystyle =\displaystyle \left(a+b\right)\left(5+v\right)Factorise

Example 2

Factorise the following expression: 2x+xz-40y-20yz

Worked Solution
Create a strategy

Group the expression in pairs and factorise using the law, A\left(C+D\right)+B\left(C+D\right)= \left(A+B\right)\left(C+D\right).

Apply the idea

To factorise an expression with 4 terms, we factorise two pairs of terms, then take out the common factor binomial.

\displaystyle 2x+xz-40y-20yz\displaystyle =\displaystyle \left(2x+xz\right) + \left(-40y -20yz\right)Group terms with common factors
\displaystyle =\displaystyle x\left(2+z\right)-20y\left(2+z\right)Factorise each pair
\displaystyle =\displaystyle \left(2+z\right)\left(x-20y\right)Factorise

Example 3

Factorise: 121m^{2}-64

Worked Solution
Create a strategy

Factorise using the difference of two squares, A^{2}-B^{2}=\left(A+B\right) \left(A-B\right)

Apply the idea

We can tell that this is a difference of two squares expression because there are 2 terms that are both perfect squares, and the sign between them is a minus (-).

Since (11m)^2=121m^2 and 8^2=64, we can use A^{2}-B^{2}=\left(A+B\right) \left(A-B\right) where A=11m and B=8.

\displaystyle 121m^{2}-64\displaystyle =\displaystyle \left(11m+8\right)\left(11m-8\right)Factorise
Idea summary

We can factorise the product of two binomial expressions using the rule: \left(A+B\right) \left(C+D\right)=AC+AD+BC+BD.

There are two special cases of factorising these expressions:

  • \left(A+B\right)^{2}=A^{2}+2AB+B^{2} (called a perfect square)

  • \left(A+B\right) \left(A-B\right)=A^{2}-B^{2} (called a difference of two squares)

Outcomes

MA5.2-6NA

simplifies algebraic fractions, and expands and factorises quadratic expressions

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