A-Level MathsCubics


The most basic cubics questions might ask you to factorise a simple cubic where a factor of x can be taken out first:

Example 1x^3+4x^2+3x=x\left(x^2+4x+3\right)=x(x+3)(x+1)
Example 2x^3-16x=x\left(x^2-16\right)=x(x+4)(x-4)
Example 32x^3+7x^2-9x=x\left(2x^2+7x-9\right)=x(2x+9)(x-1)

Other cubics questions might involve factorising a more general cubic and may require knowledge of the factor theorem.

Example 1 – Given that x=-2 is a root of the cubic x^3-x^2-x+1, factorise it completely.
Example 2 – Factorise f(x)=x^3-x^2-x+1 completely. By inspection, we can see that x=1 is a root of f(x) and so (x-1) is a factor. Using polynomial division, or inspection again, we have that f(x)=(x-1)(x^2-1) which factorises completely to f(x)=(x-1)(x-1)(x+1)=(x-1)^2(x+1).


You may also be required to sketch the graph of a given cubic.

Practice curve sketching.