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: 3x^3-2x^2+5x=x\left(3x^2-2x+5\right). Note that the discriminant of 3x^2-2x+5 is (-2)^2-4\times 3\times 5=-56 which is negative and so 3x^2-2x+5 has no roots. It follows that the original cubic cannot be factorised further.

Other cubics questions might involve factorising a more general cubic and may require knowledge of the factor theorem. See Example 2.

Sketching Cubics

  • Identify whether the cubic is positive or negative
  • Substitute x=0 to identify the y-intercept.
  • Factorise and set y=0 to identify the roots. Note that, in y=x(x+1)^2 for example, x=-1 is a repeated root and the curve must touch the x-axis at x=-1.
  • Place the graph on the axes so that all the above criteria are satisfied.

Example 1

  1. x^3+4x^2+3x
  2. x^3-16x
  3. 2x^3+7x^2-9x

Example 2

  1. Given that x=-2 is a root of the cubic x^3+x^2-x+2, factorise it completely.
  2. Factorise f(x)=x^3-x^2-x+1 completely.