FP1 2016 4 Edexcel

4

(a)

We have:

\displaystyle z = \frac {4(1 - i)} {(1 + i)(1 - i)} = \frac {4 (1 - i)} {1^2 + 1^2} = 2(1 - i) = 2 - 2i

(b)

We have:

\displaystyle z^2 = (2 - 2i)^2 = 4 - 8i -4 = -8i

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Note that non-real roots of a polynomial with real coefficients come in conjugate pairs, so 2 + 2i is also a root of the quadratic. Using vieta’s relations we have:

p = -(2 + 2i + 2 - 2i) = -4

and:

q = (2 + 2i)(2 - 2i) = 2^2 + 2^2 = 8

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