Unknown

Hints 119

7.10. Let u = r + s and v = rs. Express the quadratic factors in terms of

u and v.

7.11. Let x = gn+l and try to factor the quadratic in x. Can 320n2+144n-

243 be factored over Z?

7.12. Remember, 1+ 1 = 0. Since a repeated factor of a polynomial divides

its derivative, examine the greatest common divisor of z” + x0 + 1

and its derivative.

7.14. The following facts will be useful to keep in mind:

(1) pr = 0 for each r in Z, ;

(2) xq-x- 1 ixm- 1 iff xm - 1 E 0 (mod xq - x - 1). Thus setting

x’J = x + 1 should lead to xm - 1 - 0;

(3) (y + %)” = yp + 9;

(4) xm=x.xq. ....xqp-l and xq = x = 1, x4 = (x + 1)‘J = xQ + 1 =

x+2,...;

(5) ByFermat’sTheorem,xP-x haspzeros0,1,2,...,p-1. What

does this say about its factorization over Z,?

7.16. Let x = ty. Check when the zeros oft2+t+l are zeros of (t+l)n-tn-l

and its derivative.

7.21. If f(x,d = (X - ddd, w h a t is the relationship between q(x, y)

and dy, xl?

7.22. Look at the cases n = 2,3,4 and make a conjecture.

8.3. Observe that for any integer x, the left side lies between (x2 +~/2)~

and (x2 + x/2 + l)2.

8.6. The integers r - ai are nonzero and distinct. Arrange them in ascend-

ing order of absolute value and examine their product.

8.7. If t is an integer root, then m2 + (t2 + t)m - (t3 - 1) = 0 is solvable

for integer m. The discriminant (t2 + 3t - 4)2 + (24t - 20) must be a

perfect square not strictly between (t2 + 3t - 3)2 and (t” + 3t - 5)2.

8.8. If m is an integer zero, then f(m - ck) E 0 (mod k) for each integer

C.

8.9. Identify four linear factors of f(t) - 12. What can be said about the

factorization of the integer f(k) - 12?

8.10. What can be said about m + n + k?

8.11. Solve the system for (x-P)~ and (x-q)2; use a discriminant condition.
Compare coefficients. Guess.