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3.8. Other Problems 113

3.8 Other Problems

1. Suppose that x2 = yx - 1 and y2 = 1 - y. Show that x5 = 1 but that
   2 # 1.


2. Is

correct?

a3 + b3 a+b

a3 + (a - b)3 = a+(a-b)

3. For which integers x is x4 + x3 + x2 + z + 1 a perfect square?


4. Let a and b be rational. Can t5 - t - 1 and t2 + at + b ever have a
   common complex zero?


5. Find a and b so that the equations x3 + ax2 + 11x + 6 = 0 and
   x3 + bx2 + 14x + 8 = 0 may have two roots in common.


6. Suppose that al, a2,... , az,, are distinct integers such that

(x - al)(x - a2)(x - as)... (x - azfl) + (-1)“-‘(n!)2 = 0

has an integer root r. Show that

2nr = al + a2 + ag +... + azn.

7. Find all integer values of m for which the polynomial

t3-mt2-mt-(m2+1)

has an integer zero.

8. Prove that, if f(t) is a polynomial with integer coefficients and there
   exists a positive integer k such that none of the integers f( 1)) f(2),... ,
   f(k) is divisible by k, then f(t) has no integer zeros.


9. Let f be a manic polynomial over Z and suppose that there are four
   distinct integers a, b, c, d for which

f(a) = f(b) = f(c) = f(d) = 12.

Show that there is no integer k for which f(k) = 25.

10. Let the zeros m, n, k of t3 + at + b be rational. Prove that the zeros
    of mt2 + nt + k are rational.


11. Find constants a, b, c, d, p, q for which

a(x - P)~ + b(x - q)2 = 5x2 + 8x + 14

c(x - p)” + d(x - q)2 = x2 + 102 + 7.