(^156) 4. Equations
- Determine a manic cubic polynomial over Q one of whose zeros is
1 _ p/3 + p/3.
Hints
Chapter 41.1. Add the equations and determine x + y + z first.1.3. Use Exercise 2 and the first two equations to obtain x : y : z.1.4. Set t = u and consider the two equations as a linear system in the
“variables” u2, u, 1.1.6. (a) Multiply the first equation by xy, XZ, respectively. Treat the other
equations similarly.1.11. (a) Add the equations ai = rbi.3.3. (c) t”+2 = yP+1 - tk.3.4. Use the quadratic and cubic cases to predict what k should be.3.8. We wish to find a number p + 21f4q + 21f2r + 23f4s such that
(a+. ..)(~+a. .) is an element of F. Begin by multiplying (a+21i2c)+
2114(b + 21i2d) by a number to yield a product with no terms in 2114.
Now multiply by a second factor which disposes of the terms in 21i2.3.9. If a + b&i is a zero of f(t), then so also is a - b&. Use the Factor
Theorem to determine a quadratic factor of f(t) over F. Or, the third
zero is the sum of the zeros less 2a.6.10. If p = qr is prime, then either q or r is 1. How often can a polynomial
assume the values fl?
7.1. Write equation as difference of squares equals 0.7.2. Let u = x2 + x - 1, v = x2 - x - 2. Write as equation in u and v.
Factor.7.3. Let x = 1 -t.7.4. Write the left side as (u - a’)(~ + a2), where u = x2 + 5ax + 5a2.7.7. Square both sides and write in terms of y = x[x - (a + b)].7.9. Use the Factor Theorem to find the form of p(t) - t and also of
p(p(t)) -p(t). Note that p(p(t)) -t is the sum of these two quantities.