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258 Answers to Exercises and Solutions to Problems

a from the last two inequalities yields respectively -6 5 a - 2c 5 6 and

-5 5 b + 3c 5 5. Taking account of ICI 5 1 leads to Ial < 8 and Ibl _< 8.

Hence Ial + lb] + ICI 2 17. Equality holds for the polynomial 8x2 - 82 + 1 =

2(2x - 1)2 - 1.

9.1. a - bi is also a root; since the sum of the roots is 0, the third root is

-2~. Hence the product of the roots is -2u(u2 + b2) = -9, from which the

result follows.

9.2. Let g(z) be the constant polynomial c. Then f(c) = c, so that f(x) = x

for each x.

9.3. Observe that (u2 + b2 - 2)2 + (c2 + d2 - 2)2 + 2(uc - bd)2 =

(2 + c2 - 2)2 + (b2 + d2 - 2)2 + 2(ub - ~d)~. Each of I and II is equiv-

alent to one side of this identity vanishing.

9.4. Trying x = u + fi yields (u” - 3u2 + 5~ - 6 + ~U(U - 2)1/2 = 0, which

is evidently satisfied by u = 2. Thus, x = 2 + $ 2.

9.5. If u+b = 0, i.e. u = -b, then both equations are identities in x. On the

other hand, suppose a + b # 0. Two roots of the first equation are -a and

-b. Multiplying this equation by x yields a nontrivial quadratic equation,

so that these are the only two roots. Both of these are roots of the second

equation.

9.6. (a) Since (-u - d-)(-a + dn) = b, it follows that b/x =

-a + dn from which the result follows.

(b) The equation for y can be rewritten y + c + dn = 0, where

c = up - Q and d = bp2 - Pupq + q2. Now apply (a) to (y, c, d).

9.7. We make two initial observations:

(1) 22x - 15 - 8x2 = -(2x -3)(4x - 5) 2 0 if and only if 514 5 x 5 312.

(2) (x2 - 2x + 1) - (22x -^15 - 8x2) = (3x - 4)2 >^0 so that

1 - x + 422x - 15 - 8x2 < 0 when 514 5 x 5 312, x # 413.

Hence, the square roots of the given expression are
0 when x = 413
pure imaginary (i times a real) when 5/4 5 x 5 3/2, x # 4/3
not real, not pure imaginary otherwise.
Consider the case x < 514 or x > 312, and let the square root be u + iv.
Then
.U2-$=l-x

whence

4u2v2 = 8x2 - 22x + 15

(u2 + v2)2 = (u” - v2)2 + 4u2v2 = (3x - 4)?