# 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.

### 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.

### Cubics Exam Questions

CubicsExamQuestions

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