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Hints 117

2.8. Let g = u + v and h be two polynomials with 0 5 degv < deg u.

Show that gh cannot be homogeneous. Recall Exercise 1.7.4(d).

2.10. (a) (b) W ri ‘t e as the sum of two polynomials which have a factor in

common.

(c) (e) (f) Use the Factor Theorem.

(d) If reducible, there must be a homogeneous linear factor. Try a

symmetric one.

(g) If reducible, th ere must be a homogeneous linear factor. Then

for some a, b, the polynomial must vanish under the substitution

~=.ax+by.

2.11. Express pn(x, y,z) as a polynomial in Z. Use long division by z2 -
(x + y)z + xy; alternatively, experiment with some numerical values
of x and y to get a handle on a possible factorization.

2.12. Use the Factor Theorem to find linear factors. What is the factoriza-
tion when z = O?

2.13. What is the significance of the divisibility of the leading coefficient
by m?

2.14. Test small moduli.

3.4. (b) If a/b is a root, then the long division algorithm in which q(t) is

divided by bt - a should yield integer coefficients at every stage in

the quotient.

4.12. Note that 2t3 + t + 3 = (t + 1)(2t2 - 2t + 3).

5.12. (a) Show that C is a primitive Pkth root of unity iff c2 is a primitive
kth root.
(b) Show that (’ is a primitive kth root iff -C is a primitive 2kth root.

6.4. Use induction on k. Choose cl to express

p(t) -- Cl

q(t) t - al

in the form pl(t)/ql(t) where ql(t) has the zeros a2,... , ak.

6.7. We seek a representation of the form