Factorise sum and difference of cubes
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.
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.