116 3. Factors and Zeros- A regular polygon of seven sides is inscribed in a circle of unit radius.
Prove that the length of a side of the polygon is a root of the equation
x6 - 7x4 + 14x2 - 7 = 0
and state the geometrical significance of the other roots.- Let f(n) = n3 + 396n2 - llln + 38. Prove that f(n) s 0 (mod 3”)
has precisely nine solutions modulo 3” for all integers m 2 5. - Show that it is not possible to find polynomials f(t), g(t) over C
which are coprime such that
-s-z--. f(t+ 1) f(t)^1
dt + 1) t?(t) I!Hints
Chapter 31.5. Suppose t2 + 1 = (at + b)(ct + d). Derive a contradiction if a, b, c, d
are real.
1.8. Reducibility over Z requires care. If at2 + bt + c has rational zeros, it
can be written in the form (u/v)(pt + q)(rt + s), where gcd(u,v) =
gcd(p, q) = gcd(r, s) = 1. Must v be equal to l?1.16. Suppose h(t) is reducible and can be written as the product of Gait’
and Chit’. Consider the two cases: (1) p divides only one of a0 and bo;
(2) p divides each of a0 and bo. In case (l), compare with the proof of
the Eisenstein Criterion. In case (2), show by induction that p must
divide ok and bk for 0 5 k 5 m; look at the coefficients of t2k and tk.
1.21. Consider the greatest common divisor of f(t) and g(t). Consult Ex-
ercise 1.6.2 and Exploration E.20.
2.7. (d) Write 11 as the sum of integers of opposite parity whose product
is -24.28 = -25 a3.7.
(f) Apply the Eisenstein Criterion.
(i) Factor 3t4 - 2t3 - t2.
(j) (k) Look for an integer zero.
(n) Eliminate the possibility of a linear factor. Try the method of
undetermined coefficients.
(0) Assume a factorization (t2 + at + b)(t3 + ct2 + dt + e). Show that
b and e have opposite parity and that, in fact, b is even and e is odd.
This reduces the pair (b,e) to two possibilities, up to sign.
(p) Write as a difference of squares.