| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2016 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Find constants with divisibility condition |
| Difficulty | Standard +0.3 This is a standard Factor Theorem application requiring polynomial division or coefficient comparison to find two unknowns, followed by solving a quartic that factors nicely. The method is routine for P3 level—set up equations by equating coefficients after division, solve the linear system, then factor the resulting polynomial. Slightly above average difficulty due to the quartic and two-step process, but follows a well-practiced algorithm with no novel insight required. |
| 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 |
| Commence division by \(x^2 - x + 2\) and reach a partial quotient \(4x^2 + kx\) | M1 | |
| Obtain quotient \(4x^2 + 4x + a - 4\) or \(4x^2 + 4x + b/2\) | A1 | |
| Equate \(x\) or constant term to zero and solve for \(a\) or \(b\) | M1 | |
| Obtain \(a = 1\) | A1 | |
| Obtain \(b = -6\) | A1 | [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Show that \(x^2 - x + 2 = 0\) has no real roots | B1 | |
| Obtain roots \(\frac{1}{2}\) and \(-\frac{3}{2}\) from \(4x^2 + 4x - 3 = 0\) | B1 | [2] |
## Question 4:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Commence division by $x^2 - x + 2$ and reach a partial quotient $4x^2 + kx$ | M1 | |
| Obtain quotient $4x^2 + 4x + a - 4$ or $4x^2 + 4x + b/2$ | A1 | |
| Equate $x$ or constant term to zero and solve for $a$ or $b$ | M1 | |
| Obtain $a = 1$ | A1 | |
| Obtain $b = -6$ | A1 | [5] |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Show that $x^2 - x + 2 = 0$ has no real roots | B1 | |
| Obtain roots $\frac{1}{2}$ and $-\frac{3}{2}$ from $4x^2 + 4x - 3 = 0$ | B1 | [2] |
---4 The polynomial $4 x ^ { 4 } + a x ^ { 2 } + 11 x + b$, where $a$ and $b$ are constants, is denoted by $\mathrm { p } ( x )$. It is given that $\mathrm { p } ( x )$ is divisible by $x ^ { 2 } - x + 2$.\\
(i) Find the values of $a$ and $b$.\\
(ii) When $a$ and $b$ have these values, find the real roots of the equation $\mathrm { p } ( x ) = 0$.
\hfill \mbox{\textit{CAIE P3 2016 Q4 [7]}}