Cubics - A-Level Maths by StudyWell

Cubics

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.