242 8. Miscellaneous Problems
- (a) 1 - p(x) can be written as a product.
- We want fi + 6 + fi to be a zero of p(x) - (x - &)q(x). What
is the polynomial of smallest degree in y = x - 6 over Z with the
zero y= Ji+fi? - Two squarings with some rearranging leads to a product of two
quadratics set equal to zero. Thus the four possible roots can be
explicitly identified and analyzed. - Look at the sign of the coefficients, the discriminant and the product
(Q + b - c)(b + c - a)(c + a - b), where a, b, c are the zeros. - Look at cases according as a, b, c are positive, negative or zero. The
hard case is that in which none are zero and the signs differ. Let
2 = cos 21, etc. - A real zero can be guaranteed as the sum of the values of the poly-
nomial for two different substitutions of the variable vanishes. - Solve the problem for m = n = 2 by considering (AX - BY)3.
- Let q(x) = p(x) +.. a. What is the parity of the degree of q(x)? What
can be said about the extremum of q(x)? - Let f(t) and g(t) b e cubits with the zeros x, y, z and a, b, c respec-
tively. Look at the sign of (f - g)(t) at a and c. - Let the zeros of the polynomial be ri. Recall the factorization of
2ā - rr where n = 1 000 000. - Consider the number of distinct zeros of p(x), q(x) and (p - q)(x).
How many zeros does pā(x) have counting multiplicity? - The system p + q + r = 0, up + vq + wr = 0, xp + yq + zr =^0
has a nontrivial solution for (p, q, r). Show that bu + cx = bv + cy =
bw + cz = 1 for some b and c. Use the conditions cx = 1 - bu and
z3 = a3 - u3 to obtain a cubic equation satisfied by u. The same
equation is satisfied by v and w. - Multiply by the product of the (x-k) to render the difference between
the two sides of the inequality in the polynomial form p(x). What is
the relation between the endpoints of the interval and the zeros of
p(x)? - Show that the left side multiplied by (u + b) is symmetric in all six
variables. You will need to show that Cabc = 0. - Write a + bi, a - bi, c + di, c- di in terms of the symmetric functions
of x = cos u + i sin u, etc. Use the first three equations to solve for
the symmetric functions which can then be plugged into the fourth.