| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2022 |
| Session | October |
| Marks | 6 |
| Topic | Factor & Remainder Theorem |
| Type | Solve p(algebraic transform) = 0 |
| Difficulty | Moderate -0.3 Part (a) is a straightforward factorisation problem (factor out x, then solve a quadratic using the formula or factorisation), requiring only standard algebraic techniques. Part (b) uses a simple substitution x = (y-2)² to convert back, then requires careful consideration of which solutions are valid (only non-negative x values give real y). While part (b) adds a layer of complexity with the substitution and checking validity, this is still a routine multi-step question testing standard A-level techniques without requiring novel insight. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
In this question you should show all stages of your working.
Solutions relying on calculator technology are not acceptable.
\begin{enumerate}[label=(\alph*)]
\item Using algebra, find all solutions of the equation
$$3x^3 - 17x^2 - 6x = 0$$ [3]
\item Hence find all real solutions of
$$3(y - 2)^6 - 17(y - 2)^4 - 6(y - 2)^2 = 0$$ [3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM 2022 Q7 [6]}}