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26 1. Fundamentals

6. Let p(t) = at3 +- bt2 + ct + d be a cubic polynomial whose zeros are
   x, Y, z*
   (a) Show that

p(t) = p(t) - p(x) = (t - x)(at2 + (ax + b)t + (ax2 + bx + c)).

(b) Show that p(t) can be written in the form

a(t - x)(t - y)(t - z).

(c) By expanding the product in (b) and comparing coefficients,

verify that

x+y+z=-b/a

xy+yz+zx=c/Q

xyz = -d/a.

7. Find a necessary and sufficient condition on p, q, r that the zeros of

t3 + pt2 + qt + r

are in arithmetic progression.

8. Express each of the following polynomials as a polynomial in the
   elementary symmetric functions sr = x + y + z, s2 = xy + yz + zx,
   s3 = xyz:
   x3 + y3 + z3
   x2y3 + x3y2 + x2z3 + x3z2 + y2z3 + 9%”

(2 + Y>(Y + z)(z +x).

9. (a) Verify that

x3+y3+z3-3xyz = (x+y+z)(x2+y2+z2-xy-xz-yz).

(b) Write x2 + y2 + z2 - xy - xz - yz as the sum of three squares of

polynomials and deduce that this quantity is nonnegative when-

ever 2, y, z are real.

(c) Prove the arithmetic-geometric mean inequality: if a, b, c 2 0,

then

(abc)“3 5 (Q + b + c)/3

with equality if and only if a = b = c.