| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2007 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Two polynomials, shared factor or separate conditions |
| Difficulty | Moderate -0.8 This is a straightforward application of the Remainder Theorem and polynomial division with clear signposting. Part (a) requires simple substitution of x=3/2, part (b) uses the factor theorem to find d (answer given), and part (c) is routine polynomial division or coefficient comparison. All steps are standard textbook exercises requiring only direct application of learned techniques with no problem-solving insight needed. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02k Simplify rational expressions: factorising, cancelling, algebraic division |
| Answer | Marks | Guidance |
|---|---|---|
| \(f\!\left(\frac{3}{2}\right) = 2\!\left(\frac{3}{2}\right)^3 - 7\!\left(\frac{3}{2}\right)^2 + 13 = 4\) | M1, A1 (2 marks) | Substitute \(\pm\frac{3}{2}\) in \(f(x)\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(g\!\left(\frac{3}{2}\right) = 0 \Rightarrow d + 4 = 0 \Rightarrow d = -4\) | M1A1 (2 marks) | AG (convincingly obtained); SC written explanation with \(g\!\left(\frac{3}{2}\right)=0\); not seen/clear E2,1,0 |
| Answer | Marks | Guidance |
|---|---|---|
| \(a = -2\), \(b = -3\) | B1, B1 (2 marks) | Inspection expected. By division: M1 – complete method, A1 CAO. Multiply out and compare coefficients: M1 – evidence of use, A1 – both \(a\) and \(b\) correct |
## Question 2:
**Part (a)**
$f\!\left(\frac{3}{2}\right) = 2\!\left(\frac{3}{2}\right)^3 - 7\!\left(\frac{3}{2}\right)^2 + 13 = 4$ | M1, A1 (2 marks) | Substitute $\pm\frac{3}{2}$ in $f(x)$
**Part (b)**
$g\!\left(\frac{3}{2}\right) = 0 \Rightarrow d + 4 = 0 \Rightarrow d = -4$ | M1A1 (2 marks) | AG (convincingly obtained); SC written explanation with $g\!\left(\frac{3}{2}\right)=0$; not seen/clear E2,1,0
**Part (c)**
$a = -2$, $b = -3$ | B1, B1 (2 marks) | Inspection expected. By division: M1 – complete method, A1 CAO. Multiply out and compare coefficients: M1 – evidence of use, A1 – both $a$ and $b$ correct
**Total: 6 marks**
---2 The polynomial $\mathrm { f } ( x )$ is defined by $\mathrm { f } ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } + 13$.
\begin{enumerate}[label=(\alph*)]
\item Use the Remainder Theorem to find the remainder when $\mathrm { f } ( x )$ is divided by $( 2 x - 3 )$.
\item The polynomial $\mathrm { g } ( x )$ is defined by $\mathrm { g } ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } + 13 + d$, where $d$ is a constant.
Given that ( $2 x - 3$ ) is a factor of $\mathrm { g } ( x )$, show that $d = - 4$.
\item Express $\mathrm { g } ( x )$ in the form $( 2 x - 3 ) \left( x ^ { 2 } + a x + b \right)$.
\end{enumerate}
\hfill \mbox{\textit{AQA C4 2007 Q2 [6]}}