| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2022 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Prove root count with given polynomial |
| Difficulty | Moderate -0.8 This is a highly structured, routine question testing basic factor theorem application, polynomial division, and discriminant analysis. Each part guides students through standard procedures with no problem-solving required—part (a) is simple substitution, (b) is algebraic expansion/comparison, (c) uses the discriminant formula, and (d) is a simple transformation. This is easier than average A-level content. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(f(-3) = 2(-3)^3 + 5(-3)^2 + 2(-3) + 15 = -54 + 45 - 6 + 15\) | M1 | Attempts \(f(-3)\); attempted division by \((x+3)\) or \(f(3)\) is M0; look for evidence of embedded values or two correct terms |
| \(f(-3) = 0 \Rightarrow (x+3)\) is a factor | A1 | Must achieve and state \(f(-3)=0\) with suitable conclusion; sight of \(f(x)=0\) when \(x=-3\) acceptable; must follow M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| At least 2 of: \(a=2,\ b=-1,\ c=5\) | M1 | Correct method implied by values for at least 2 correct constants; allow embedded in \(f(x)\) or within working |
| All of: \(a=2,\ b=-1,\ c=5\) | A1 | All values correct; isw incorrectly stated values of \(a\), \(b\), \(c\) following correct quadratic |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(b^2 - 4ac = (-1)^2 - 4(2)(5)\) | M1 | Either: considers discriminant using their \(a,b,c\); or attempts completing the square \(2\!\left(x \pm \tfrac{1}{4}\right)^2+\ldots\); or attempts quadratic formula; or sketches U-shaped quadratic not crossing \(x\)-axis |
| \(b^2-4ac = -39 < 0\) so quadratic has no real roots, so \(f(x)=0\) has only 1 real root | A1 | Correct calculation; explanation that quadratic has no real roots; minimal conclusion that \(f(x)=0\) has only one root |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((x =)\ 2\) | B1 | Condone \((2,\ 0)\) |
## Question 2:
### Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $f(-3) = 2(-3)^3 + 5(-3)^2 + 2(-3) + 15 = -54 + 45 - 6 + 15$ | M1 | Attempts $f(-3)$; attempted division by $(x+3)$ or $f(3)$ is M0; look for evidence of embedded values or two correct terms |
| $f(-3) = 0 \Rightarrow (x+3)$ is a factor | A1 | Must achieve and state $f(-3)=0$ with suitable conclusion; sight of $f(x)=0$ when $x=-3$ acceptable; must follow M1 |
### Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| At least 2 of: $a=2,\ b=-1,\ c=5$ | M1 | Correct method implied by values for at least 2 correct constants; allow embedded in $f(x)$ or within working |
| All of: $a=2,\ b=-1,\ c=5$ | A1 | All values correct; isw incorrectly stated values of $a$, $b$, $c$ following correct quadratic |
### Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $b^2 - 4ac = (-1)^2 - 4(2)(5)$ | M1 | Either: considers discriminant using their $a,b,c$; or attempts completing the square $2\!\left(x \pm \tfrac{1}{4}\right)^2+\ldots$; or attempts quadratic formula; or sketches U-shaped quadratic not crossing $x$-axis |
| $b^2-4ac = -39 < 0$ so quadratic has no real roots, so $f(x)=0$ has only 1 real root | A1 | Correct calculation; explanation that quadratic has no real roots; minimal conclusion that $f(x)=0$ has only one root |
### Part (d)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(x =)\ 2$ | B1 | Condone $(2,\ 0)$ |
---2.
$$f ( x ) = 2 x ^ { 3 } + 5 x ^ { 2 } + 2 x + 15$$
\begin{enumerate}[label=(\alph*)]
\item Use the factor theorem to show that $( x + 3 )$ is a factor of $\mathrm { f } ( x )$.
\item Find the constants $a$, $b$ and $c$ such that
$$f ( x ) = ( x + 3 ) \left( a x ^ { 2 } + b x + c \right)$$
\item Hence show that $\mathrm { f } ( x ) = 0$ has only one real root.
\item Write down the real root of the equation $\mathrm { f } ( x - 5 ) = 0$
\end{enumerate}
\hfill \mbox{\textit{Edexcel AS Paper 1 2022 Q2 [7]}}