Review question

# Can we sketch the reciprocal of a polynomial? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R6299

## Question

Express $x^3 - 3x - 2$ as the product of three linear factors.

Show that $x^3 - 3x - 2$ may also be expressed in the form $(x-1)^2(x+p) + q$, where $p$ and $q$ are constants to be found. Hence, or otherwise, show that the curve whose equation is $y = x^3 - 3x - 2$ has a minimum point where $x = 1$.

Sketch the curve whose equation is $\begin{equation*} y = \frac{1}{x^3 - 3x - 2}, \end{equation*}$ indicating clearly all its asymptotes. Hence obtain the number of real roots of the equation $\begin{equation*} (x+1)(x^3 - 3x - 2) = 1. \end{equation*}$