| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2011 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Finding Constants from Factor or Zero Remainder Conditions |
| Difficulty | Moderate -0.3 This is a straightforward application of the factor theorem and polynomial division. Part (i) requires setting up two simultaneous equations using p(1)=0 and p(-2)=0, which is routine. Part (ii) involves recognizing that x²+x-2=(x-1)(x+2) and performing polynomial division, which is a standard technique. The question requires no novel insight and follows predictable patterns, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Substitute \(x = 1\) or \(x = -2\) and equate to zero | M1 | |
| Obtain a correct equation in any form with powers of \(x\) values calculated | A1 | |
| Obtain a second correct equation in any form | A1 | |
| Solve a relevant pair of equations for \(a\) or for \(b\) | M1 | |
| Obtain \(a = 3\) and \(b = -5\) | A1 | [5] |
| (ii) Attempt division by \(x^2 + x - 2\), or equivalent, and reach a partial quotient of \(x^2 + kx\) | M1 | |
| Obtain partial quotient \(x^2 + 2x\) | A1 | |
| Obtain \(x^2 + 2x - 1\) with no errors seen | A1 | |
| S.C. M1A1√ if '\(a\)' and/or '\(b\)' incorrect | [3] |
**(i)** Substitute $x = 1$ or $x = -2$ and equate to zero | M1 |
Obtain a correct equation in any form with powers of $x$ values calculated | A1 |
Obtain a second correct equation in any form | A1 |
Solve a relevant pair of equations for $a$ or for $b$ | M1 |
Obtain $a = 3$ and $b = -5$ | A1 | [5]
**(ii)** Attempt division by $x^2 + x - 2$, or equivalent, and reach a partial quotient of $x^2 + kx$ | M1 |
Obtain partial quotient $x^2 + 2x$ | A1 |
Obtain $x^2 + 2x - 1$ with no errors seen | A1 |
S.C. M1A1√ if '$a$' and/or '$b$' incorrect | | [3]6 (i) The polynomial $x ^ { 4 } + a x ^ { 3 } - x ^ { 2 } + b x + 2$, where $a$ and $b$ are constants, is denoted by $\mathrm { p } ( x )$. It is given that $( x - 1 )$ and $( x + 2 )$ are factors of $\mathrm { p } ( x )$. Find the values of $a$ and $b$.\\
(ii) When $a$ and $b$ have these values, find the quotient when $\mathrm { p } ( x )$ is divided by $x ^ { 2 } + x - 2$.
\hfill \mbox{\textit{CAIE P2 2011 Q6 [8]}}