| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2005 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Single polynomial, two remainder/factor conditions |
| Difficulty | Moderate -0.8 This is a straightforward application of the Remainder Theorem requiring students to substitute x=2 and x=-1 to create two simultaneous equations, then solve for a and b. Part (b) is verification by substitution. Standard C2 textbook exercise with routine algebraic manipulation and no problem-solving insight required. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| solving \(\begin{cases}2a + b = 1\\-a + b = 31\end{cases} \Rightarrow a = -10, b = 21\) | M1 A1, M1 A1, M1 A1 | (6 marks) |
| (b) \(f(3) = 27 - 18 + 3a + b = 27 - 18 - 30 + 21 = 0\) | \(\therefore (x-3)\) is a factor | M1, A1 c.s.o |
**(a)** $f(2) = 1 \Rightarrow 8 - 2 \times 4 + 2a + b = 1$
$f(-1) = 28 \Rightarrow -1 - 2 - a + b = 28$
solving $\begin{cases}2a + b = 1\\-a + b = 31\end{cases} \Rightarrow a = -10, b = 21$ | M1 A1, M1 A1, M1 A1 | (6 marks)
**(b)** $f(3) = 27 - 18 + 3a + b = 27 - 18 - 30 + 21 = 0$ | $\therefore (x-3)$ is a factor | M1, A1 c.s.o | (2 marks)
**(Subtotal: 8 marks)**
Guidance: (a) 1st two M marks: attempting $f(\pm 2)$ and $f(\pm 1)$. A1 A1: for each correct, unsimplified equation. 3rd M1: for solving two linear equations $\to a =$ or $b=$. A1: both values.
(b) M1: Attempting $f(3)$. A1: $= 0$ with comment.\begin{enumerate}
\item $\quad \mathrm { f } ( x ) = x ^ { 3 } - 2 x ^ { 2 } + a x + b$, where $a$ and $b$ are constants.
\end{enumerate}
When $\mathrm { f } ( x )$ is divided by ( $x - 2$ ), the remainder is 1 .\\
When $\mathrm { f } ( x )$ is divided by $( x + 1 )$, the remainder is 28 .\\
(a) Find the value of $a$ and the value of $b$.\\
(b) Show that ( $x - 3$ ) is a factor of $\mathrm { f } ( x )$.\\
\hfill \mbox{\textit{Edexcel C2 2005 Q5 [8]}}