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Hints^47

(i) i = cos(?r/2) + isin(?r/2), so that

i(cos 4 + i sin 4) = cos(+ + x/2) + i sin(d + 7r/2);

(ii) cos 4 = sin(d + x/2) and sin 4 = - cos(d + x/2).

Ultimately, obtain expressions for the roots of the equation in terms

of e = 412 + x/4.

8.18. Assume that the factorization over Z occurs. Show that the discrim-
inants are perfect squares of the same parity. Find u and v such
that b2 = u2 + v2. To get lb1 as a sum of squares, use the equation
(r + is)2 = u + iv as inspiration.

8.19. Put the expression in the form A + Bcos20 + Csin20 = A+
D sin(20 + 4).

8.20. If f(x, y) h as 1 inear factors, the discriminant of f(x, y) as a quadratic
in x must be square as a quadratic in y; what can be said about the
discriminant of the second quadratic?

8.21. (a) Substituting y = mx+c into the other equation yields a quadratic

equation with equal roots.

(b) Use the theory of the quadratic to determine the midpoints of the

chords. If 1(x, y) = 0 is on the locus, find m and c in terms of x and

y, and substitute into the condition obtained in (a).

9.1. What are the other two roots? Do not solve the equation; just use

the fact that the coefficients are real and one of them is 0.

9.2. In particular, f commutes with any constant polynomial.

9.3. A, B, C all vanish iff A2 + B2 + C2 = 0.

9.4. The observation (fi)” - fi = fi suggests trying x = fi + u.

9.6. (a) With the help of a surd conjugate, determine l/x.

(b) Simplify the left side of the first equation.

9.7. Can the square root ever be real? pure imaginary (i.e. a real multiplied

by i)?

9.9. What is xu? yv?

9.10. Expand the left side as the product of two quadratics whose leading

and linear coefficients agree, then put it in the form (u - l)(u + 1).

9.11. Use the arithmetic-geometric mean inequality (Exercise 5.9).