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114 3. Factors and Zeros

12. Determine the range of w(w + x)(w + y)(w + 2) where z, ‘y, z, w are
    real numbers which satisfy


13. If 2 + y + z = 0, prove that

x5 + y5 + %5 x3 + y3 + %3 x2 + y2 + %2

=

5 3 * 2 ’

14. If
    by/z + cr/y = a
    m/x + ax/z = b
    ax/y + by/x = c,
    show that

(a) xs3 + ym3 + C3 + x-ly-iz-l = 0

(b) a3x3 + b3y3 + c3z3 + abcxyz = 0

(c) a3 + b3 + c3 = 5abc.

15. If x + y + z = 0, show that

{

y-z

z+

z-x x-y

-+- = 9.

Y % >(

-+&+L

Y--z X-Y >

16. If ~(1 - mzy/x3) = y(1 - mxz/$) = ~(1 - myx/,$) with x, y, t
    unequal, prove that each quantity is equal to x + y + t - m.


17. Suppose a, b, c, d are integers, that r is a zero of P(x) = x3 + ax2 +
    bx - 1, r + 1 is a zero of y3 + cy2 + dy + 1, and that P(x) is irreducible
    over Q. Express another zero s of P(x) as a function of r which does
    not explicitly involve a, b, c or d.


18. Let n = 2m, where m is an odd integer greater than 1. Let 0 =
    cos(2a/n) + i sin(2x/n). Express (1 - ~9)~’ explicitly as a polynomial
    in 8, i.e. akfl” +Uk-iflk-’ +... + al 0 + ac, with integer coefficients ai.


19. Let F be a finite field having an odd number m of elements. Let p(x)
    and q(x) be irreducible polynomials over F of the form x2 + bx + c.

(a) Prove that q(z) = p(x + h) - k for some h and k in F.

(b) For how many elements k in F is p(x) + k irreducible?

20. Show that the product of four consecutive terms of an arithmetic
    progression of integers plus the fourth power of the common difference
    is a perfect square. Give a nontrivial example in which this quantity
    is actually a perfect fourth power.