| Exam Board | SPS |
|---|---|
| Module | SPS SM Statistics (SPS SM Statistics) |
| Year | 2022 |
| Session | February |
| Marks | 7 |
| Topic | Factor & Remainder Theorem |
| Type | Prove root count with given polynomial |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question testing basic Factor Theorem application, polynomial division (or expansion and comparison), and discriminant analysis. All steps are routine textbook exercises with no problem-solving insight required, making it easier than average but not trivial since it requires executing multiple standard techniques correctly. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02k Simplify rational expressions: factorising, cancelling, algebraic division |
1.
Answer all the questions.
$$f ( x ) = 3 x ^ { 3 } - 7 x ^ { 2 } + 7 x - 10$$
\begin{enumerate}[label=(\alph*)]
\item Use the factor theorem to show that ( $x - 2$ ) is a factor of $\mathrm { f } ( x )$
\item Find the values of the constants $a , b$ and $c$ such that
$$\mathrm { f } ( x ) \equiv ( x - 2 ) \left( a x ^ { 2 } + b x + c \right)$$
\item Using your answer to part (b) show that the equation $\mathrm { f } ( x ) = 0$ has only one real root.\\[0pt]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Statistics 2022 Q1 [7]}}