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94 3. Factors and Zeros

9. We can check whether a is a zero of a polynomial q(t) over Z by
   evaluating q(a) by Horner’s Method. In the case q(t) = 4t3 - 22t2 +
   7t + 15, a = 5, we get the table for Horner’s and Newton’s Methods
   respectively:

4 -22 7 15 15 7 -22 4

20 -10 -15 3 2 -4

4 -2 -3 0 15 10 -20 0

Is there any connection between these tables? Explain.

10. In using Newton’s Table to check a negative rational zero, what dif-
    ference does it make to consider the numerator to be negative and
    the denominator to be positive rather than the other way around?


11. Consider the linear polynomial ct + d. If a/b is a zero of this, then
    a 1 d and b 1 c. Consider rationals which satisfy these conditions, and
    explain what happens in the bottom line of the table for Newton’s
    Method of Divisors for the unsuccessful candidates.


12. Let 0 = 2~15.

(a) Verify that cos tJ/2 + cos 28 = 0.

(b) Show that 2~ = cos 0 satisfies the equation

x = 2(4x4 - 42’ + 1) - 1.

(c) Factor 8x4 - 8x2 - x + 1 over Z, and deduce that cos 0 is a zero

of a quadratic polynomial over Z.

(d) Determine cos 0.

Exploration

E.31. Find all integers n for which the zeros of the quadratic polynomial

nt2 + (n + 1)t - (n + 2)

are rational.

3.4 Locating Integer Roots: Modular Arithmetic

What is a quick way to see that the polynomial t2 - 131t + 267 cannot
possibly have an integer zero? Such a zero must be even or odd. If t is
given an even value, then t2 and -1332 are even, and the polynomial must