Question 9 - A-Level Maths

Edexcel AS Paper 1 2018 June — Question 9 9 marks

Exam Board	Edexcel
Module	AS Paper 1 (AS Paper 1)
Year	2018
Session	June
Marks	9
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Factor & Remainder Theorem
Type	Express as product with specific form
Difficulty	Standard +0.3 This is a straightforward multi-part question on factor theorem requiring routine algebraic manipulation. Part (a) is simple substitution, part (b) involves polynomial division and recognizing a perfect square (guided by the given form), and part (c) applies the factorized form to solve inequalities—all standard AS-level techniques with clear scaffolding.
Spec	1.02j Manipulate polynomials: expanding, factorising, division, factor theorem 1.02n Sketch curves: simple equations including polynomials

9. $$g ( x ) = 4 x ^ { 3 } - 12 x ^ { 2 } - 15 x + 50$$

Use the factor theorem to show that $( x + 2 )$ is a factor of $\mathrm { g } ( x )$.
Hence show that $\mathrm { g } ( x )$ can be written in the form $\mathrm { g } ( x ) = ( x + 2 ) ( a x + b ) ^ { 2 }$, where $a$ and $b$ are integers to be found. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f7935caa-6626-4ba8-87ef-e9bb59e1ac3e-22_517_807_607_621} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve with equation $y = \mathrm { g } ( x )$
Use your answer to part (b), and the sketch, to deduce the values of $x$ for which
1. $\mathrm { g } ( x ) \leqslant 0$
2. $\mathrm { g } ( 2 x ) = 0$

Show mark scheme Show mark scheme source

Question 9(a):

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
$g(-2) = 4\times(-2)^3 - 12\times(-2)^2 - 15\times(-2) + 50$ embedded	M1	1.1b — Attempts $g(-2)$, some sight of $(-2)$ embedded. Any attempt to divide or factorise is M0
$g(-2) = 0 \Rightarrow (x+2)$ is a factor	A1	2.4 — Requires correct statement AND conclusion. Both "$g(-2)=0$" and "$(x+2)$ is a factor" must be seen

Question 9(b):

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
$4x^3 - 12x^2 - 15x + 50 = (x+2)(4x^2 - 20x + 25)$	M1, A1	1.1b — Attempts to divide $g(x)$ by $(x+2)$. If inspection: expect $(x+2)(4x^2......\pm 25)$
$= (x+2)(2x-5)^2$	M1, A1	1.1b — Attempts to factorise $(4x^2 - 20x + 25)$, usual rule $(ax+b)(cx+d)$, $ac=\pm4$, $bd=\pm25$. $(x+2)(-2x+5)^2$ also correct

Question 9(c)(i):

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
$x \leqslant -2,\ x = 2.5$	M1, A1ft	1.1b — For identifying solution where curve is on or below axis. Follow through on their $g(x)=(x+2)(ax+b)^2$ only where $ab < 0$

Question 9(c)(ii):

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
$x = -1,\ x = 1.25$	B1ft	2.2a — For deducing solutions of $g(2x)=0$ will be where $x=-1$ and $x=1.25$. Condone coordinates $(-1,0)$ and $(1.25,0)$

## Question 9(a):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $g(-2) = 4\times(-2)^3 - 12\times(-2)^2 - 15\times(-2) + 50$ embedded | M1 | 1.1b — Attempts $g(-2)$, some sight of $(-2)$ embedded. Any attempt to divide or factorise is M0 |
| $g(-2) = 0 \Rightarrow (x+2)$ is a factor | A1 | 2.4 — Requires correct statement AND conclusion. Both "$g(-2)=0$" and "$(x+2)$ is a factor" must be seen |

---

## Question 9(b):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $4x^3 - 12x^2 - 15x + 50 = (x+2)(4x^2 - 20x + 25)$ | M1, A1 | 1.1b — Attempts to divide $g(x)$ by $(x+2)$. If inspection: expect $(x+2)(4x^2......\pm 25)$ |
| $= (x+2)(2x-5)^2$ | M1, A1 | 1.1b — Attempts to factorise $(4x^2 - 20x + 25)$, usual rule $(ax+b)(cx+d)$, $ac=\pm4$, $bd=\pm25$. $(x+2)(-2x+5)^2$ also correct |

---

## Question 9(c)(i):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $x \leqslant -2,\ x = 2.5$ | M1, A1ft | 1.1b — For identifying solution where curve is on or below axis. Follow through on their $g(x)=(x+2)(ax+b)^2$ only where $ab < 0$ |

## Question 9(c)(ii):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $x = -1,\ x = 1.25$ | B1ft | 2.2a — For deducing solutions of $g(2x)=0$ will be where $x=-1$ and $x=1.25$. Condone coordinates $(-1,0)$ and $(1.25,0)$ |

---

Show LaTeX source

9.

$$g ( x ) = 4 x ^ { 3 } - 12 x ^ { 2 } - 15 x + 50$$
\begin{enumerate}[label=(\alph*)]
\item Use the factor theorem to show that $( x + 2 )$ is a factor of $\mathrm { g } ( x )$.
\item Hence show that $\mathrm { g } ( x )$ can be written in the form $\mathrm { g } ( x ) = ( x + 2 ) ( a x + b ) ^ { 2 }$, where $a$ and $b$ are integers to be found.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{f7935caa-6626-4ba8-87ef-e9bb59e1ac3e-22_517_807_607_621}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}

Figure 2 shows a sketch of part of the curve with equation $y = \mathrm { g } ( x )$
\item Use your answer to part (b), and the sketch, to deduce the values of $x$ for which
\begin{enumerate}[label=(\roman*)]
\item $\mathrm { g } ( x ) \leqslant 0$
\item $\mathrm { g } ( 2 x ) = 0$
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{Edexcel AS Paper 1 2018 Q9 [9]}}

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