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Complex roots of real polynomials

mathematical

Let

p(z) = z^4 - 4z^3 + 14z^2 - 36z + 45

The roots 3i, 2 + i, 2 − i and −3i on an Argand diagram, joined to form a trapezium with parallel vertical sides of lengths 6 and 2, a distance 2 apart.

Facts you may need

Powers of $i$ : $i^2 = -1$ , $i^3 = -i$ , $i^4 = 1$ — For substituting $3i$.
Conjugate pair $a \pm bi$ gives the factor $z^2 - 2az + (a^2 + b^2)$ — For $\pm3i$: $z^2 + 9$.
Trapezium area $= \frac12(a + b)h$ — Parallel sides $a$, $b$, distance $h$ apart.

(a)

Show that

z = 3i

is a root of

p(z) = 0

(b)

Hence solve

p(z) = 0

fully.

(c)

The four roots are plotted on an Argand diagram and joined to form a quadrilateral. Find its area.