| Exam Board | Edexcel |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2022 |
| Session | June |
| Marks | 3 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Single unknown constant |
| Difficulty | Moderate -0.8 This is a straightforward application of the factor theorem requiring students to substitute x = -2 into f(x), set equal to zero, and solve for k. It's a single-step problem with routine algebraic manipulation, making it easier than average but not trivial since it requires careful handling of the expanded form. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Sets \(f(-2) = 0 \Rightarrow (-2-4)((-2)^2 - 3\times-2 + k) - 42 = 0\) | M1 | Attempts \(f(-2)=0\) leading to equation in \(k\). Condone slips but expect first bracket of \((-2-4)\). "\(-42\)" must not be omitted but could appear as \(+42\) with sign slip. May expand \(f(x)=(x-4)(x^2-3x+k)-42\) first — condone slips but 42 must be present. FYI expanded: \(f(x)=x^3-7x^2+(12+k)x-4k-42\) |
| \(-6(k+10) = 42 \Rightarrow k = \ldots\) | M1 | Solves a linear equation in \(k\) from setting \(f(\pm2)=0\). The \(\pm42\) must appear at substitution. Allow minimal evidence. If \(f(x)\) expanded, dependent on cubic containing \(kx\) term and a '42' |
| \(k = -17\) | A1 | Correct answer following correct work; allow recovery from invisible brackets |
## Question 2:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Sets $f(-2) = 0 \Rightarrow (-2-4)((-2)^2 - 3\times-2 + k) - 42 = 0$ | M1 | Attempts $f(-2)=0$ leading to equation in $k$. Condone slips but expect first bracket of $(-2-4)$. "$-42$" must not be omitted but could appear as $+42$ with sign slip. May expand $f(x)=(x-4)(x^2-3x+k)-42$ first — condone slips but 42 must be present. FYI expanded: $f(x)=x^3-7x^2+(12+k)x-4k-42$ |
| $-6(k+10) = 42 \Rightarrow k = \ldots$ | M1 | Solves a **linear** equation in $k$ from setting $f(\pm2)=0$. The $\pm42$ must appear at substitution. Allow minimal evidence. If $f(x)$ expanded, dependent on cubic containing $kx$ term and a '42' |
| $k = -17$ | A1 | Correct answer following correct work; allow recovery from invisible brackets |
---\begin{enumerate}
\item $\mathrm { f } ( x ) = ( x - 4 ) \left( x ^ { 2 } - 3 x + k \right) - 42$ where $k$ is a constant Given that $( x + 2 )$ is a factor of $\mathrm { f } ( x )$, find the value of $k$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel Paper 1 2022 Q2 [3]}}