| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2024 |
| Session | October |
| Marks | 9 |
| Topic | Factor & Remainder Theorem |
| Type | Show equation reduces to polynomial |
| Difficulty | Moderate -0.3 This is a slightly below-average A-level question. Part (i) involves routine factor theorem verification and polynomial factorization—standard C2 content. Part (ii) requires logarithm manipulation using standard laws to arrive at the given polynomial form, then applying domain restrictions to identify the valid root. While it connects two topics (polynomials and logarithms), each step follows predictable techniques without requiring novel insight or extended problem-solving. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.06f Laws of logarithms: addition, subtraction, power rules1.06g Equations with exponentials: solve a^x = b |
In this question you must show detailed reasoning.
The polynomial f(x) is given by
$$f(x) = x^3 + 6x^2 + x - 4.$$
\begin{enumerate}[label=(\roman*)]
\item \begin{enumerate}[label=(\alph*)]
\item Show that $(x + 1)$ is a factor of f(x). [1]
\item Hence find the exact roots of the equation f(x) = 0. [4]
\end{enumerate}
\item \begin{enumerate}[label=(\alph*)]
\item Show that the equation
$$2\log_2(x + 3) + \log_2 x - \log_2(4x + 2) = 1$$
can be written in the form f(x) = 0. [3]
\item Explain why the equation
$$2\log_2(x + 3) + \log_2 x - \log_2(4x + 2) = 1$$
has only one real root and state the exact value of this root. [1]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM 2024 Q9 [9]}}