Factorise sum and difference of cubes

Lesson

Practice

Lesson

We've looked at how to factorise monic and non-monic quadratics. We've also looked at some special factorising rules, such as the difference of two squares.

In this chapter we are going to look at how to factorise the sum and difference of two cubics (ie. terms with powers of $3$3).

Let's start by looking at the general forms of these rules.

General Forms for Factorising Cubics

Sum of two cubics: $a^3+b^3=\left(a+b\right)\left(a^2-ab+b^2\right)$a3+b3=(a+b)(a2−ab+b2)

Difference of two cubics: $a^3-b^3=\left(a-b\right)\left(a^2+ab+b^2\right)$a3−b3=(a−b)(a2+ab+b2)

Some people use the mnemonic "SOAP" to help remember the order of the signs in these formulae. The letters stand for:

SAME as the sign in the middle of the original expression

OPPOSITE sign to the original expression

ALWAYS POSITIVE

Now let's look at some examples and see this process in action!

Worked Examples

Question 1

Factorise $x^3+27$x3+27.

Question 2

Factorise $x^3-1000$x3−1000.

Question 3

Evaluate $5^3-11^3$53−113 by factorising it first.

Note: You must show your factorisation.

Question 4

Simplify $\frac{125x^3+8}{5x+2}$125x3+85x+2.

Outcomes

9.A.P.3

Recall of algebraic expressions and identities. Further identities of the type: (x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx, (x ± y)^ 3 = x^3 ± y^3 ± 3xy (x ± y), x^3 + y^3 + z^3 – 3xyz = (x + y + z) (x^2 + y^2 + z^2 – xy – yz – zx) and their use in factorization of polynomials. Simple expressions reducible to these polynomials.