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4.4. The Fundamental Theorem of Algebra 141

8. Solve graphically the equations:

(a) z3 - 72 + 6 = 0

(b) .z3 - t - 1 = 0

(c) 23 + i = 0.

9. (a) Show that th e t ransformation z = x + yi applied to the equation

%4 - 2%2 + 3% - 2 = 0

yields the simultaneous real equations

x4 - 6x2$ + y4 - 2x2 + 2y2 + 3x - 2 = 0

423y - 4x$ - 4xy + 3y = 0.

(b) Show that (C) can be rewritten in the form

y2 = (3x2 - 1) f d8z4 - 4x2 - 3x + 3.

((3

CD)

When -2 < x < 1, show that there is one positive value for y2

giving two real values of y. When x < -2 or x > 1, show that

there are two distinct positive values of y2 giving four real values

of y. When z = -2, and x = 1, show that there are two values

of y2, one of which is zero, and that there are three values of y.

(c) Find tan 0, tan2 0, tan 30, tan2 38 for 0 = 7r/8, and determine

the asymptotes of the curves (C) and (D).

(d) Sketch the loci of the equations (C) and (D), and on the same

axes indicate a circle with center at the origin and radius suffi-

ciently large that the interior of the circle contains all the inter-

section points of the curves (C) and (D). Label all the intersec-

tion points of the circle and the locus of (C) with R and of the

circle and the locus of (D) with I. Verify that the points R and

I alternate.

10. Let n be a positive integer. Consider the polynomial equation

Zn + a,-1%+-l + Q,-2z”-2 + * f f + al% + al-J = 0.

Suppose that z = x+ya’ transforms this equation to u(x, y)+iv(x, y) =

0, where u and v are polynomials over R.

We wish to examine where a circle with centre at the origin and a very

large radius r intersects each of the loci u(x, y) = 0 and v(x, y) = 0.

Let (r cose,rsin8) be a typical point on this circle. Use de Moivre’s

Theorem (Exercise 1.3.8) to show that

u(r cos 0, r sin 0) = r” cos n0 + terms of lower degree in T