| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2023 |
| Session | September |
| Marks | 9 |
| Topic | Factor & Remainder Theorem |
| Type | Two polynomials, shared factor or separate conditions |
| Difficulty | Standard +0.3 This is a straightforward application of the Factor Theorem requiring students to substitute x=3 into both polynomials, set equal to zero, and solve simultaneous equations. The factorization in part (b) is routine once p and q are found. All steps are standard textbook procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
7.\\
$( x - 3 )$ is a common factor of $\mathrm { f } ( x )$ and $\mathrm { g } ( x )$ where:
$$\begin{aligned}
& \mathrm { f } ( x ) = 2 x ^ { 3 } - 11 x ^ { 2 } + ( p - 15 ) x + q \\
& \mathrm {~g} ( x ) = 2 x ^ { 3 } - 17 x ^ { 2 } + p x + 2 q
\end{aligned}$$
\begin{enumerate}[label=(\alph*)]
\item (i) Show that $3 p + q = 90$ and $3 p + 2 q = 99$
Fully justify your answer.\\
(a) (ii) Hence find the values of $p$ and $q$.
\item $\quad \mathrm { h } ( x ) = \mathrm { f } ( x ) + \mathrm { g } ( x )$
Using your values of $p$ and $q$, fully factorise $\mathrm { h } ( x )$
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q7 [9]}}