# Cubics

## Factorising Cubics

The most basic cubics questions might ask you to factorise a simple cubic where a factor of x can be taken out first. For instance, the terms in the expression have a common factor of x and so factor x out to give . You might think that this can be factorised further, however, in this case the quadratic cannot be factorised. This can be seen by noting that the discriminant of is which is negative and so has no roots. – see Discriminants. It follows that the cubic cannot be factorised further. In most cases, however, there will be some more factoring required. See Example 1.

In addition to the above, other cubics questions might involve factorising a more general cubic and may require knowledge of the factor theorem. See Example 2.

## Sketching Cubics

- Firstly, identify whether the cubic is positive or negative.
- Then, substitute into the cubic expression to identify the -intercept.
- Next, factorise if possible and set to identify the roots. Note that, in for example, is a repeated root and the curve must touch the x-axis at .
- Finally, place the graph on the axes so that all the above criteria are satisfied.

## Examples

## Videos

Factorising and sketching a simple cubic with a little coordinate geometry thrown in.

Sketching and transforming a cubic mixed with a little differentiation.

## Extra Resources

The following PDF has 9 cubic sketching exercises to complete. You will find the solutions to the exercises on the second page.

Open Cubic Sketching Exercise in New Window

CubicsHere are the Cubics Questions by Topic. You will find solutions to these past exam questions at the back of the document.

Open Cubics Questions in New Window

CubicsExamQuestions