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44 1. Fundamentals

15. How many distinct terms are there in the expansion of

2&l + x2)(21 + x2 + 23) *. * (Xl + x2 +.. * + zn)?

16. Let u be an integer. Simplify


17. Show that there are infinitely many pairs of positive integers m and
    n for which 4mn - m - n +^1 is a perfect square.


18. Determine all numbers u for which

(i) there is a cubic polynomial p with integer coefficients for which

u, u2, u3 are distinct zeros;

(ii) u is nonrational.

19. For any polynomial p(t) = a,P + am-ltm-l + ... + alt + a~, let
    r(p(t))=a~+a~_l+...+Q:+ag.
    Let f(t) = 3t2 + 7t + 2. Find, with proof, a polynomial g(t) for which

6) s(O) = 1;

(ii) r(f(t)“) = r(g(t)“) for n = 1,2,....

20. Given that x2 + # = 6ry and z > y > 0, determine

x+Y -.

X-Y

Hints

Chapter 1

1.12. (a) The constant term is the value of the polynomial at 0.

(b) The difference of the two polynomials is identically zero.

1.13. (a) deg f(2t) = degf(t). What is deg h(i)?

1.14. 1og2t = log2 + 1ogt.

1.15. g(t + k) -g(t) is identically equal to 0.

1.17. p(f - g) is identically zero.

1.18. Either use induction or multiply both sides by 1 -t.