| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2017 |
| Session | March |
| Marks | 10 |
| 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 requiring verification of a given root and then polynomial division to find remaining roots. While it involves complex numbers (indicated by variable z), the mechanical process of substitution and factorization is routine for Further Maths students, making it slightly easier than average. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02i Quadratic equations: with complex roots |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Substitute \(z = -1+\mathrm{i}\) and attempt expansions of the \(z^2\) and \(z^4\) terms | M1 | |
| Use \(\mathrm{i}^2 = -1\) at least once | M1 | |
| Complete the verification correctly | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State second root \(z = -1-\mathrm{i}\) | B1 | |
| Carry out complete method for finding quadratic factor with zeros \(-1+\mathrm{i}\) and \(-1-\mathrm{i}\) | M1 | |
| Obtain \(z^2 + 2z + 2\), or equivalent | A1 | |
| Attempt division of \(p(z)\) by \(z^2+2z+2\) and reach partial quotient \(z^2+kz\) | M1 | |
| Obtain quadratic factor \(z^2 - 2z + 5\) | A1 | |
| Solve 3-term quadratic and use \(\mathrm{i}^2 = -1\) | M1 | |
| Obtain roots \(1+2\mathrm{i}\) and \(1-2\mathrm{i}\) | A1 |
## Question 8(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Substitute $z = -1+\mathrm{i}$ and attempt expansions of the $z^2$ and $z^4$ terms | M1 | |
| Use $\mathrm{i}^2 = -1$ at least once | M1 | |
| Complete the verification correctly | A1 | |
**Total: 3**
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## Question 8(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| State second root $z = -1-\mathrm{i}$ | B1 | |
| Carry out complete method for finding quadratic factor with zeros $-1+\mathrm{i}$ and $-1-\mathrm{i}$ | M1 | |
| Obtain $z^2 + 2z + 2$, or equivalent | A1 | |
| Attempt division of $p(z)$ by $z^2+2z+2$ and reach partial quotient $z^2+kz$ | M1 | |
| Obtain quadratic factor $z^2 - 2z + 5$ | A1 | |
| Solve 3-term quadratic and use $\mathrm{i}^2 = -1$ | M1 | |
| Obtain roots $1+2\mathrm{i}$ and $1-2\mathrm{i}$ | A1 | |
**Total: 7**
---(i) Showing all your working, verify that $u$ is a root of the equation $\mathrm { p } ( z ) = 0$.\\
(ii) Find the other three roots of the equation $\mathrm { p } ( z ) = 0$.\\
\hfill \mbox{\textit{CAIE P3 2017 Q8 [10]}}