| Exam Board | AQA |
|---|---|
| Module | AS Paper 2 (AS Paper 2) |
| Year | 2022 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Factorise polynomial completely |
| Difficulty | Moderate -0.8 This is a straightforward application of the factor theorem requiring verification of f(3)=0, identification of a sign error in Statement 2 (should be x-3, not x+3), and routine polynomial division to complete the factorisation. The question tests basic understanding rather than problem-solving skills, making it easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(f(3) = 54 - 63 - 36 + 45 = 0\), Kaya is correct that \(f(3)=0\) | E1 | AO 2.4 — shows 3 substituted and \(f(3)=0\) stated |
| But Kaya's conclusion is wrong; \((x-3)\) is a factor of \(f(x)\) | E1 | AO 2.3 — states Kaya incorrect and gives correct conclusion. Or states \(f(-3) \neq 0\). Or states \((x+3)\) is not a factor |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((2x^3 - 7x^2 - 12x + 45) \div (x-3) = 2x^2 - x - 15\) | M1 | AO 1.1a — divides \(f(x)\) by \((x-3)\) or uses inspection; must obtain \(2x^2\) or \(15\) |
| \((x-3)^2(2x+5)\) — correct quadratic factor and both remaining roots of \(f(x)=0\) | A1 | AO 1.1b |
| Fully correct factorisation of \(f(x)\) | A1 | AO 1.1b — ISW |
## Question 5(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $f(3) = 54 - 63 - 36 + 45 = 0$, Kaya is correct that $f(3)=0$ | E1 | AO 2.4 — shows 3 substituted and $f(3)=0$ stated |
| But Kaya's conclusion is wrong; $(x-3)$ is a factor of $f(x)$ | E1 | AO 2.3 — states Kaya incorrect and gives correct conclusion. Or states $f(-3) \neq 0$. Or states $(x+3)$ is not a factor |
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## Question 5(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(2x^3 - 7x^2 - 12x + 45) \div (x-3) = 2x^2 - x - 15$ | M1 | AO 1.1a — divides $f(x)$ by $(x-3)$ or uses inspection; must obtain $2x^2$ or $15$ |
| $(x-3)^2(2x+5)$ — correct quadratic factor and **both** remaining roots of $f(x)=0$ | A1 | AO 1.1b |
| Fully correct factorisation of $f(x)$ | A1 | AO 1.1b — ISW |
---5 Kaya is investigating the function
$$f ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } - 12 x + 45$$
Kaya makes two statements.\\
Statement 1: $\mathrm { f } ( 3 ) = 0$\\
Statement 2: this shows that ( $x + 3$ ) must be a factor of $\mathrm { f } ( x )$.\\
5
\begin{enumerate}[label=(\alph*)]
\item State, with a reason, whether each of Kaya's statements is correct.
Statement 1: $\_\_\_\_$\\
Statement 2: $\_\_\_\_$\\
5
\item Fully factorise f (x).
\end{enumerate}
\hfill \mbox{\textit{AQA AS Paper 2 2022 Q5 [5]}}