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16 1. Fundamentals

(a) z3 = 1

(b) z4 = 1

(c) 26 = 1

(d) zd= 1.

Indicate the solutions of each equation on an Argand diagram.

10. (a) Let a,b be real numbers. Find real numbers z and y for which
    (x + yi)2 = a + hi.
    (b) Determine the square roots of -7 - 24i.


11. Solve the equations

(a) t2 + 3t + 3 - i = 0

(b) t2 + (2i - 1)t + (5i + 1) = 0.

12. Show that every quadratic equation with complex coefficients has at
    least one complex root, and therefore can be written as the product
    of two linear factors with complex coefficients.


13. Prove that 11 + izl = II- izl if and only if z is real.


14. (a) Let p(t) b e a polynomial with real coefficients. Show that, for
    any complex number w, p(E) = p(w). Deduce that, if w is a zero
    of p(t), then so is Z.
    (b) Give a counterexample to show that (a) is not true in general if
    p(t) has some nonreal coefficients.


15. Let n be a nonnegative integer. The Tchebychef Polynomial Tn(z) is
    defined, for -1 5 x 5 1, by

To(x) = I

T,,(x) = cos n(arc cos z) (n^2 1).

(a) Show that Tn+l(c) - 2xTn(z) + Tn-l(x) = 0 (n > 1).

(b) Find TI(x), TV, T3(x) and T4(x). Sketch the graphs of these

functions.

(c) For each n, establish that T,(x) is a polynomial and determine

its degree. (Do this in two ways, by (a) and by de Moivre’s

Theorem.)

(d) Show that

G(x) = ‘2(-y ( ; ) xn-2yl _ x2)t.

Td