| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2014 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Prove root count with given polynomial |
| Difficulty | Moderate -0.3 This is a straightforward application of the factor theorem and polynomial division. Part (i) requires substituting x=-2 and solving for a (routine). Part (ii) involves polynomial division to get a quadratic, then checking its discriminant is negative - all standard techniques with no novel insight required. Slightly easier than average due to being a textbook-style multi-part question with clear signposting. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks |
|---|---|
| (i) Substitute \(-2\) and equate to zero, or divide and equate remainder to zero | M1 |
| Obtain \(a = 12\) | A1 [2] |
| (ii) Carry out division, or equivalent, at least as far as \(x^2\) and \(x\) terms in quotient | M1 |
| Obtain \(x^2 - 2x + 6\) | A1 |
| Calculate discriminant of a 3 term quadratic quotient (or equivalent) | DM1 |
| Obtain \(-20\) (or equivalent) | A1 |
| Conclude by referring to, or implying, root \(-2\) and no root from quadratic factor | A1 [5] |
(i) Substitute $-2$ and equate to zero, or divide and equate remainder to zero | M1 |
Obtain $a = 12$ | A1 [2] |
(ii) Carry out division, or equivalent, at least as far as $x^2$ and $x$ terms in quotient | M1 |
Obtain $x^2 - 2x + 6$ | A1 |
Calculate discriminant of a 3 term quadratic quotient (or equivalent) | DM1 |
Obtain $-20$ (or equivalent) | A1 |
Conclude by referring to, or implying, root $-2$ and no root from quadratic factor | A1 [5] |6 The polynomial $\mathrm { p } ( x )$ is defined by
$$\mathrm { p } ( x ) = x ^ { 3 } + 2 x + a$$
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, find the quotient when $\mathrm { p } ( x )$ is divided by ( $x + 2$ ) and hence show that the equation $\mathrm { p } ( x ) = 0$ has exactly one real root.
\hfill \mbox{\textit{CAIE P2 2014 Q6 [7]}}