Unknown

6.1. Interpreting the Coefficients of a Polynomial 195

(b) Prove that, if x, y, z are real, then a2 >3b and that z (and, by

symmetry, x and y as well) must lie in the closed interval [u, v],

where

u= (l/3) (- a -2&G)

21 = (l/3) (-a + 2JKG)

Note that p’(u) = p’(v) = o2 - 3b.

(c) With the hypotheses of (b), prove that p(u) 5 0 5 p(v) and

deduce that

]2a3 - 9ab + 27~1 5 2(a2 - 3b)3/2

and thence that

a2b2 - 4a3c + 18abc - 4b3 - 27c2 2 0.

(d) Suppose that a2b2 - 4a3c + 18abc - 4b3 - 27c2 > 0. Show that

a2 - 3b 2 0 and that the cubic p(t) has three real zeros.

[Cf. Problems 1.4.4, 5.4.15, 6.2.5.1

10. Solve the equation t4 - t3 - 7t2 + 23t - 20 = 0, given that the product
    of two of the roots is -5.


11. Consider the polynomial equation

x4+px3+qx2+rx+s=0.

(a) Prove that the product of two of its roots is equal to the product

of the other two iff r2 = p2s.

(b) Prove that, if the sum of two of its roots is equal to the sum of

the other two, then p” + 8r = 4pq.

(c) Suppose that p3 + 8r = 4pq. Must the sum of two of its roots

equal the sum of the other two?

12. In Exercise 1.4.11, the quartic equation

t4 + pt2 + qt + r = 0... (1)

was solved by factoring the left side as (t2 + ut + v)(t2 - ut + w),

where u satisfies the equation

t6 + 2pt4 + (p2 - 4r)t2 - q2 = 0.... (2)

(a) Argue that the roots of (2) can be given as f2a, f2b, f2c, where

a, b, c are selected so that 8abc = -q.