| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2012 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Solve p(algebraic transform) = 0 |
| Difficulty | Standard +0.3 This is a straightforward multi-part question on the factor theorem. Part (i) requires simple substitution to find a constant. Part (ii)(a) is routine factorization after finding one factor. Part (ii)(b) involves a standard algebraic transformation p(x²)=0, requiring students to solve x²=r for each root r—a common exam technique but slightly above routine drill. Overall slightly easier than average due to clear structure and standard methods. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Substitute \(x = 2\) and equate to zero, or divide by \(x - 2\) and equate constant remainder to zero, or equivalent | M1 | |
| Obtain \(a = 4\) | A1 | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Find further (quadratic or linear) factor by division, inspection or factor theorem or equivalent | M1 | |
| Obtain \(x^2 + 2x - 8\) or \(x + 4\) | A1 | |
| State \((x-2)^2(x+4)\) or equivalent | A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State any two of the four (or six) roots | B1\(\checkmark\) | |
| State all roots \((\pm\sqrt{2},\ \pm 2i)\), provided two are purely imaginary | B1\(\checkmark\) | [2] |
## Question 3:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Substitute $x = 2$ and equate to zero, or divide by $x - 2$ and equate constant remainder to zero, or equivalent | M1 | |
| Obtain $a = 4$ | A1 | [2] |
### Part (ii)(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Find further (quadratic or linear) factor by division, inspection or factor theorem or equivalent | M1 | |
| Obtain $x^2 + 2x - 8$ or $x + 4$ | A1 | |
| State $(x-2)^2(x+4)$ or equivalent | A1 | [3] |
### Part (ii)(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State any two of the four (or six) roots | B1$\checkmark$ | |
| State all roots $(\pm\sqrt{2},\ \pm 2i)$, provided two are purely imaginary | B1$\checkmark$ | [2] |
---3 The polynomial $\mathrm { p } ( x )$ is defined by
$$\mathrm { p } ( x ) = x ^ { 3 } - 3 a x + 4 a$$
where $a$ is a constant.\\
(i) Given that $( x - 2 )$ is a factor of $\mathrm { p } ( x )$, find the value of $a$.\\
(ii) When $a$ has this value,
\begin{enumerate}[label=(\alph*)]
\item factorise $\mathrm { p } ( x )$ completely,
\item find all the roots of the equation $\mathrm { p } \left( x ^ { 2 } \right) = 0$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2012 Q3 [7]}}