| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2020 |
| Session | March |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Factorisation After Division or Remainder |
| Difficulty | Moderate -0.8 This is a straightforward polynomial division question with a direct 'hence' part. Part (a) is routine algebraic manipulation requiring polynomial long division or inspection. Part (b) simply requires recognizing that the remainder being 18 means the equation factors as (x²+5x+6)(quotient)=18, leading to standard quadratic solving. No novel insight needed, just methodical application of standard techniques. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Carry out division as far as \(4x + k\) | M1 | |
| Obtain quotient \(4x - 3\) | A1 | |
| Confirm remainder is \(18\) | A1 | AG necessary detail needed |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply equation is \((4x-3)(x^2+5x+6)=0\) | B1FT | Following *their* quotient from part (a) |
| Attempt solution of cubic equation to find three real roots | M1 | |
| Obtain \(-3,\ -2,\ \frac{3}{4}\) | A1 | |
| Total | 3 |
## Question 2(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Carry out division as far as $4x + k$ | M1 | |
| Obtain quotient $4x - 3$ | A1 | |
| Confirm remainder is $18$ | A1 | AG necessary detail needed |
| **Total** | **3** | |
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## Question 2(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply equation is $(4x-3)(x^2+5x+6)=0$ | B1FT | Following *their* quotient from part **(a)** |
| Attempt solution of cubic equation to find three real roots | M1 | |
| Obtain $-3,\ -2,\ \frac{3}{4}$ | A1 | |
| **Total** | **3** | |2
\begin{enumerate}[label=(\alph*)]
\item Find the quotient when $4 x ^ { 3 } + 17 x ^ { 2 } + 9 x$ is divided by $x ^ { 2 } + 5 x + 6$, and show that the remainder is 18 .
\item Hence solve the equation $4 x ^ { 3 } + 17 x ^ { 2 } + 9 x - 18 = 0$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2020 Q2 [6]}}