| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2021 |
| Session | October |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Prove root count with given polynomial |
| Difficulty | Moderate -0.3 This is a structured multi-part question testing standard Factor and Remainder Theorem techniques. Part (a) requires recognizing that substituting x=3/2 gives the remainder directly. Part (b) is routine verification by substitution. Parts (c)(i-ii) involve straightforward factorization and analyzing roots of a quadratic. While it requires multiple steps, each individual step follows textbook procedures with no novel insight needed, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| States \(-21\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Attempts \(f(3) = (3^2 - 2)(2\times3 - 3) - 21\) | M1 | Condone slips but don't accept just \(f(3)=0\). Attempts via long division score M0 |
| Achieves \(f(3)=0 \Rightarrow (x-3)\) is a factor of \(f(x)\) | A1* | If conclusion given in preamble, accept minimal conclusion such as // or QED |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Expands \(f(x) = (x^2-2)(2x-3)-21\) to correct 4 term cubic | B1 | Allow if cubic seen anywhere |
| \(f(x) = 2x^3 - 3x^2 - 4x - 15 = (x-3)(2x^2 \ldots \pm 5)\) | M1 | Correct attempt to find quadratic factor by division by \((x-3)\) or inspection; two correct terms implies the mark |
| \(= (x-3)(2x^2+3x+5)\) | A1 | Must be seen together on one line following correct cubic |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Attempts \(b^2 - 4ac\) for their \(2x^2+3x+5\) | M1 | Accept via quadratic formula, calculator, or completing the square. Factorisation attempts are M0 |
| Achieves \(b^2-4ac < 0\) and states only root is \(x=3\) | A1* | Requires correct factorisation, correct calculation, reason and conclusion. E.g. "\(3^2 - 4\times2\times5 < 0\) so \(f(x)=0\) only has root at \(x=3\)". Do not accept "only real root is \((x-3)\)" |
## Question 4:
### Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| States $-21$ | B1 | |
### Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Attempts $f(3) = (3^2 - 2)(2\times3 - 3) - 21$ | M1 | Condone slips but don't accept just $f(3)=0$. Attempts via long division score M0 |
| Achieves $f(3)=0 \Rightarrow (x-3)$ is a factor of $f(x)$ | A1* | If conclusion given in preamble, accept minimal conclusion such as // or QED |
### Part (c)(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Expands $f(x) = (x^2-2)(2x-3)-21$ to correct 4 term cubic | B1 | Allow if cubic seen anywhere |
| $f(x) = 2x^3 - 3x^2 - 4x - 15 = (x-3)(2x^2 \ldots \pm 5)$ | M1 | Correct attempt to find quadratic factor by division by $(x-3)$ or inspection; two correct terms implies the mark |
| $= (x-3)(2x^2+3x+5)$ | A1 | Must be seen together on one line following correct cubic |
### Part (c)(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Attempts $b^2 - 4ac$ for their $2x^2+3x+5$ | M1 | Accept via quadratic formula, calculator, or completing the square. Factorisation attempts are M0 |
| Achieves $b^2-4ac < 0$ and states only root is $x=3$ | A1* | Requires correct factorisation, correct calculation, reason and conclusion. E.g. "$3^2 - 4\times2\times5 < 0$ so $f(x)=0$ only has root at $x=3$". Do not accept "only real root is $(x-3)$" |
---4.
$$f ( x ) = \left( x ^ { 2 } - 2 \right) ( 2 x - 3 ) - 21$$
\begin{enumerate}[label=(\alph*)]
\item State the value of the remainder when $\mathrm { f } ( x )$ is divided by ( $2 x - 3$ )
\item Use the factor theorem to show that $( x - 3 )$ is a factor of $\mathrm { f } ( x )$
\item Hence,
\begin{enumerate}[label=(\roman*)]
\item factorise $\mathrm { f } ( x )$
\item show that the equation $\mathrm { f } ( x ) = 0$ has only one real root.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Edexcel P2 2021 Q4 [8]}}