| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2010 |
| Session | November |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Factor theorem and finding roots |
| Difficulty | Standard +0.3 This is a straightforward application of the factor theorem to find m, followed by routine polynomial factorization and solving a composite equation. The extension to p(z²)=0 requires recognizing that if α is a root of p(z)=0, then ±√α are roots of p(z²)=0, which is a standard technique. All steps are mechanical with no novel insight required, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem4.02i Quadratic equations: with complex roots |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Attempt to solve for \(m\) the equation \(p(-2) = 0\) or equivalent | M1 | |
| Obtain \(m = 6\) | A1 | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt \(p(z) \div (z + 2)\), equate a constant remainder to zero and solve for \(m\). | M1 | |
| Obtain \(m = 6\) | A1 | |
| (ii) (a) State \(z = -2\) | B1 | |
| Attempt to find quadratic factor by inspection, division, identity, ... | M1 | |
| Obtain \(z^2 + 4z + 16\) | A1 | |
| Use correct method to solve a 3-term quadratic equation | M1 | |
| Obtain \(-2 \pm 2\sqrt{3}i\) or equivalent | A1 | [5] |
| (ii) (b) State or imply that square roots of answers from part (ii)(a) needed | M1 | |
| Obtain \(\pm i\sqrt{2}\) | A1 | |
| Attempt to find square root of a further root in the form \(x + iy\) or in polar form | M1 | |
| Obtain \(a^2 - b^2 = -2\) and \(ab = (\pm)\sqrt{3}\) following their answer to part (ii)(a) | A1 | |
| Solve for \(a\) and \(b\) | M1 | |
| Obtain \(\pm\left(1 + i\sqrt{3}\right)\) and \(\pm\left(1 - i\sqrt{3}\right)\) | A1 | [6] |
**(i)** Attempt to solve for $m$ the equation $p(-2) = 0$ or equivalent | M1 |
Obtain $m = 6$ | A1 | [2]
**Alternative:**
Attempt $p(z) \div (z + 2)$, equate a constant remainder to zero and solve for $m$. | M1 |
Obtain $m = 6$ | A1 |
**(ii) (a)** State $z = -2$ | B1 |
Attempt to find quadratic factor by inspection, division, identity, ... | M1 |
Obtain $z^2 + 4z + 16$ | A1 |
Use correct method to solve a 3-term quadratic equation | M1 |
Obtain $-2 \pm 2\sqrt{3}i$ or equivalent | A1 | [5]
**(ii) (b)** State or imply that square roots of answers from part (ii)(a) needed | M1 |
Obtain $\pm i\sqrt{2}$ | A1 |
Attempt to find square root of a further root in the form $x + iy$ or in polar form | M1 |
Obtain $a^2 - b^2 = -2$ and $ab = (\pm)\sqrt{3}$ following their answer to part (ii)(a) | A1 |
Solve for $a$ and $b$ | M1 |
Obtain $\pm\left(1 + i\sqrt{3}\right)$ and $\pm\left(1 - i\sqrt{3}\right)$ | A1 | [6]10 The polynomial $\mathrm { p } ( z )$ is defined by
$$\mathrm { p } ( z ) = z ^ { 3 } + m z ^ { 2 } + 24 z + 32$$
where $m$ is a constant. It is given that $( z + 2 )$ is a factor of $\mathrm { p } ( z )$.\\
(i) Find the value of $m$.\\
(ii) Hence, showing all your working, find
\begin{enumerate}[label=(\alph*)]
\item the three roots of the equation $\mathrm { p } ( z ) = 0$,
\item the six roots of the equation $\mathrm { p } \left( z ^ { 2 } \right) = 0$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2010 Q10 [13]}}