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.

Examples

Example 1

Factorise x^2-12x+36

Worked Solution

Create a strategy

Using the perfect squares rule \left(A+B\right)^2=A^2+2AB+B^2.

Apply the idea

We know that we can use the perfect squares rule because half of -12 squared is (-6)^2=36.

For the rule, A=x and B=-6.

\displaystyle x^2-12x+36

\displaystyle =

\displaystyle \left(x-6\right)^2

Use the rule

Reflect and check

Once again, we can check the answer by expanding it. It's always worth checking if an expression fits the pattern A^2+2AB+B^2, because then we can use this special rule. We call this perfect square factorisation.

Example 2

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

Example 3

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

Worked Solution

Create a strategy

Identify the common factor between the two terms and then factorise using the rule:

AB+AC=A\left(B+C\right)

Apply the idea

We can see that \left(a+b\right) is a common factor of both terms.

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

\displaystyle =

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

Use the rule for factoring

Example 4

Factorise the following expression by grouping in pairs: 2x+xz-40y-20yz

Worked Solution

Create a strategy

To factorise an expression with four terms first group the expression in pairs so that each pair has a common factor.

Apply the idea

We can then factorise both pairs. The x is a common factor of \left(2x+xz\right) and 20y is a common factor of \left(40y-20yz\right).

\displaystyle 2x+xz-40y-20yz	\displaystyle =	\displaystyle x\left(2+z\right)-20y\left(2+z\right)	Factorise by x and 20y
	\displaystyle =	\displaystyle \left (2 + z\right)\left(x-20y\right)	Simplify

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 expanding binomials:

\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

ACMNA233

Expand binomial products and factorise monic quadratic expressions using a variety of strategies

1.07 Factorising binomial products

Ideas

Factorise binomial products

Examples

Example 1

Example 2

Example 3

Example 4

Outcomes

ACMNA233

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