2007 H2 Mathematics

Paper 1 Question 7

The polynomial $P (z)$ has real coefficients. The equation $P (z) = 0$ has a root $r e^{i θ},$ where $r > 0$ and $0 < θ < π .$

(a)

Write down a second root in terms of $r$ and $t h e t a,$ and hence show that a quadratic factor of $P (z)$ is $z^{2} - 2 r z \cos θ + r^{2} .$

[3]

(b)

Out of syllabus (roots of unity).

[4]

(c)

Out of syllabus (requires previous part).

[3]