| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Multiple unknowns with derivative condition |
| Difficulty | Standard +0.3 This is a straightforward application of the factor theorem with a derivative condition. Students substitute x=2 into both p(x) and p'(x) to get two simultaneous equations in a and b, then solve. The factorisation in part (b) follows immediately. While it requires understanding that a factor of both p(x) and p'(x) means a repeated root, the algebraic manipulation is routine for A-level, making this slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.07i Differentiate x^n: for rational n and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Substitute \(x=2\), equate to zero | M1 | Or divide by \(x-2\) and equate constant remainder to zero |
| Obtain a correct equation e.g. \(8a-40+2b+8=0\) | A1 | Seen or implied in subsequent work |
| Differentiate \(p(x)\), substitute \(x=2\) and equate result to zero | M1 | Or divide by \(x-2\) and equate constant remainder to zero |
| Obtain \(12a-40+b=0\), or equivalent | A1 | SOI in subsequent work |
| Obtain \(a=3\) and \(b=4\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply \((x-2)^2\) is a factor | M1 | |
| \(p(x)=(x-2)^2(ax+2)\) | A1 | |
| Obtain an equation in \(b\) | M1 | |
| e.g. by comparing coefficients of \(x\): \(b=4a-8\) | A1 | |
| Obtain \(a=3\) and \(b=4\) | A1 | SC: If uses \(x=-2\) in both equations allow M1 and allow A1 for \(a=-3\), \(b=-4\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Attempt division by \((x-2)\) | M1 | M1 earned if division reaches partial quotient of \(ax^2+kx\), or if inspection has unknown factor \(ax^2+ex+f\) and equation in \(e\) and/or \(f\), where \(a\) has value found in part 5(a) |
| Obtain quadratic factor \(3x^2-4x-4\) | A1 | |
| Obtain factorisation \((3x+2)(x-2)(x-2)\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply \((x-2)^2\) is a factor | B1 | |
| Attempt division by \((x-2)^2\), reaching quotient \(ax+k\) or use inspection with unknown factor \(cx+d\) reaching value for \(c\) or \(d\) | M1 | |
| Obtain factorisation \((3x+2)(x-2)^2\) | A1 | Accept \(3\left(x+\frac{2}{3}\right)(x-2)^2\) |
## Question 5(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Substitute $x=2$, equate to zero | M1 | Or divide by $x-2$ and equate constant remainder to zero |
| Obtain a correct equation e.g. $8a-40+2b+8=0$ | A1 | Seen or implied in subsequent work |
| Differentiate $p(x)$, substitute $x=2$ and equate result to zero | M1 | Or divide by $x-2$ and equate constant remainder to zero |
| Obtain $12a-40+b=0$, or equivalent | A1 | SOI in subsequent work |
| Obtain $a=3$ and $b=4$ | A1 | |
**Alternative method:**
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply $(x-2)^2$ is a factor | M1 | |
| $p(x)=(x-2)^2(ax+2)$ | A1 | |
| Obtain an equation in $b$ | M1 | |
| e.g. by comparing coefficients of $x$: $b=4a-8$ | A1 | |
| Obtain $a=3$ and $b=4$ | A1 | SC: If uses $x=-2$ in both equations allow **M1** and allow **A1** for $a=-3$, $b=-4$ |
**Total: 5 marks**
---
## Question 5(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Attempt division by $(x-2)$ | M1 | M1 earned if division reaches partial quotient of $ax^2+kx$, or if inspection has unknown factor $ax^2+ex+f$ and equation in $e$ and/or $f$, where $a$ has value found in part **5(a)** |
| Obtain quadratic factor $3x^2-4x-4$ | A1 | |
| Obtain factorisation $(3x+2)(x-2)(x-2)$ | A1 | |
**Alternative method:**
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply $(x-2)^2$ is a factor | B1 | |
| Attempt division by $(x-2)^2$, reaching quotient $ax+k$ or use inspection with unknown factor $cx+d$ reaching value for $c$ or $d$ | M1 | |
| Obtain factorisation $(3x+2)(x-2)^2$ | A1 | Accept $3\left(x+\frac{2}{3}\right)(x-2)^2$ |
**Total: 3 marks**5 The polynomial $a x ^ { 3 } - 10 x ^ { 2 } + b x + 8$, where $a$ and $b$ are constants, is denoted by $\mathrm { p } ( x )$. It is given that $( x - 2 )$ is a factor of both $\mathrm { p } ( x )$ and $\mathrm { p } ^ { \prime } ( x )$.
\begin{enumerate}[label=(\alph*)]
\item Find the values of $a$ and $b$.
\item When $a$ and $b$ have these values, factorise $\mathrm { p } ( x )$ completely.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2022 Q5 [8]}}