| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2021 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Two unknowns with show-that step |
| Difficulty | Moderate -0.3 This is a straightforward application of the Remainder Theorem requiring students to substitute values and solve simultaneous equations. The 'show that' in part (a) guides students to one equation, and part (b) is routine algebra. While it involves two unknowns, the method is standard textbook fare with no conceptual challenges beyond direct application of the theorem. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(f(-1) = (-1)^4 + a(-1)^3 - 3(-1)^2 + b(-1) + 5 = 4\) | M1 | Attempts to substitute \(\pm 1\) into \(f(x)\) and set equal to 4; condone invisible brackets on powers |
| \(1 - a - 3 - b + 5 = 4 \Rightarrow a + b = -1\) | A1* | Rearranges with no errors, achieves given answer with at least one intermediate line shown |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(f(2) = (2)^4 + a(2)^3 - 3(2)^2 + b(2) + 5 = -23\) | M1 | Attempts to substitute \(\pm 2\) into \(f(x)\) and set equal to \(\pm 23\) |
| \(8a + 2b = -32\) (e.g. \(4a + b = -16\)) | A1 | Powers should be evaluated; need not be fully gathered |
| \(b = -1 - a \Rightarrow 4a - 1 - a = -16 \Rightarrow a = \ldots\) | dM1 | Attempts to solve simultaneously, achieving a value for \(a\) or \(b\); dependent on previous M mark |
| \(a = -5, b = 4\) | A1 | cao |
# Question 1:
## Part (a):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $f(-1) = (-1)^4 + a(-1)^3 - 3(-1)^2 + b(-1) + 5 = 4$ | M1 | Attempts to substitute $\pm 1$ into $f(x)$ and set equal to 4; condone invisible brackets on powers |
| $1 - a - 3 - b + 5 = 4 \Rightarrow a + b = -1$ | A1* | Rearranges with no errors, achieves given answer with at least one intermediate line shown |
## Part (b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $f(2) = (2)^4 + a(2)^3 - 3(2)^2 + b(2) + 5 = -23$ | M1 | Attempts to substitute $\pm 2$ into $f(x)$ and set equal to $\pm 23$ |
| $8a + 2b = -32$ (e.g. $4a + b = -16$) | A1 | Powers should be evaluated; need not be fully gathered |
| $b = -1 - a \Rightarrow 4a - 1 - a = -16 \Rightarrow a = \ldots$ | dM1 | Attempts to solve simultaneously, achieving a value for $a$ or $b$; dependent on previous M mark |
| $a = -5, b = 4$ | A1 | cao |
---1.
$$f ( x ) = x ^ { 4 } + a x ^ { 3 } - 3 x ^ { 2 } + b x + 5$$
where $a$ and $b$ are constants.\\
When $\mathrm { f } ( x )$ is divided by ( $x + 1$ ), the remainder is 4
\begin{enumerate}[label=(\alph*)]
\item Show that $a + b = - 1$
When $\mathrm { f } ( x )$ is divided by ( $x - 2$ ), the remainder is - 23
\item Find the value of $a$ and the value of $b$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel P2 2021 Q1 [6]}}