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112 3. Factors and Zeros

13. (a) Determine a necessary and sufficient condition on the integers b
    and c such that both the polynomials

t2+bt+c and t2+bt+c+1

are reducible over Z..

(b) Determine integers b and c such that both the polynomials

3t2 + bt + c and 3t2 + bt + c + 1

are reducible over Z.

14. Let n be a positive integer, p be a prime, q = p” and m =
    (qp - l)/(q - 1). Prove that x’J - z - 1 divides xm - 1 over Z, if
    and only if n = 1.


15. Factor over Z: (x4 - 1)” - x - 1.


16. Prove that the polynomial (x + y)” - z” - y” is divisible by

(a) x2 + ty+ y2 where n s 5 (mod 6)

(b) (x2 + xy + y2)2 when n E 1 (mod 6)

(c) z3 + 2z2y + 2xy2 + y3 when n is a prime exceeding 3.

17. For each positive integer k, show that t5 + 1 is a factor of

(t4 - 1)(t3 - t2 + t - l)k + (t + 1)t4”-1.

18. If a positive integer m has a prime factor greater than 3, show that
    4m - 2m + 1 is composite.


19. Determine all values of the positive integer n for which 4” + n4 is
    prime.


20. Suppose that m is a positive odd integer exceeding 3. Prove that

22m + 1

5

is a composite integer.

21. Let f(x, y) b e a symmetric polynomial. Show that, if (x - y) is a
    factor of j(z, y), then so is (x - Y)~.


22. Prove that, for any positive integer n exceeding 1, the equation
    1 f 2x + 3x2 +... + nx”-1 = n2 has a rational root between 1 and 2.