3.2. Strategies for Factoring Polynomials over Z 87
(0) 15 - 3t4 - t2 - 4t + 14
(p) t” - 98t4 + 1.- Let f be a nonzero homogeneous polynomial of several variables.
Prove that, if f is reducible, then each of its factors must be homo-
geneous. - Let f be a symmetric polynomial of several variables. If f is reducible,
must each of its factors be symmetric? Justify your answer. - Factor each of the following polynomials.
(a) 5(a - b)’ - 4a2 + 4ab
(b) x2y - ys - x2z + y2z
(c) x2y - xy2 + y2z - yz2 + z2x - 2x2
(d) x3 + y3 + z3 - 3xy.z
(e) bc(b - c) + ca(c - a) + ab(a - b)
(f) x(y” - z”) + y(z2 - x2) + 2(x2 - y2)
(g) Y4Y + z) + 4x + z> + XY(X + Y)- Let n be a nonnegative integer and define the polynomial
pn(x, y, z) = xn(z - y) + y”(x - z) + z”(y - x).(a) Verify that p,(x, y, z) = 0 if any two of x, y, z are equal.
(b) Deduce from (a) that po(x, y, z) and pl(x, y, z) are the zero poly-
nomials.
(c) If n 1 2, show thatPn(X, Y, 2) = (2 - Y)(Y - z)(z - 4Qn(X, Y, z)where qn(x, y, z) is a homogeneous symmetric polynomial of de-
gree n - 2. Show that the coefficients of x”-‘, y”-‘, z”-’ in
q,,(x, y,z) are each 1.
(d) Identify qn(x, y, z) for n = 2,3,4.
(e) It is possible to get a shadow of a possible factorization by set-
ting one of the variables equal to 0 and factoring the resultant
polynomial of fewer variables. Factor p,(x, y, 0).
(f) Factor P~(x, Y, z) and p6(? Y, z).- Factor(x+y+z)k-xk-yk-zkfork=3,5,7.
- Is it true in general that if a polynomial with integer coefficients is
irreducible over Z, for some m, it must be irreducible over Z?