Review question

# Can we show this polynomial has roots $1$, $\cos\theta$ and $\sin\theta$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8923

## Question

1. Show, with working, that $\begin{equation} x^3 - (1 + \cos\theta + \sin\theta)x^2 + (\cos\theta\sin\theta + \cos\theta + \sin\theta)x - \sin\theta\cos\theta, \qquad \label{eq:1} \end{equation}$

equals $(x - 1)(x^2 - (\cos\theta + \sin\theta)x + \cos\theta\sin\theta).$ Deduce that the cubic in $\eqref{eq:1}$ has roots $1, \qquad \cos\theta, \qquad \sin\theta.$

2. Give the roots when $\theta = \dfrac{\pi}{3}$.

3. Find all values of $\theta$ in the range $0 \leq \theta < 2\pi$ such that two of the three roots are equal.

4. What is the greatest possible difference between two of the roots, and for what values of $\theta$ in the range $0 \leq \theta < 2\pi$ does this greatest difference occur?

Show that for each such $\theta$ the cubic $\eqref{eq:1}$ is the same.