Unknown

3.2. Strategies for Factoring Polynomials over Z 85

6t + u and vt - 20 are both integer multiples of the same linear

polynomial. Write the given polynomial in the form

6t2 + ui! + vt - 20

and thence factor it.

(b) For the quadratic at2 + bt + c, show how the determination of

integers u and v with u + v = b and uv = ac can lead to a

factorization of the quadratic into linear factors over Z.

(c) Factor each of the following quadratics and determine the values

of t for which it is negative:

28t2 + 57t + 14

20t2 + 39t - 44.

2. (a) Consider the polynomial ak - bk. Use the Factor Theorem to de-
   duce that a-b is a factor, and thence write the given polynomial
   as the product of two factors.
   (b) Let k be an odd integer. Use the Factor Theorem (Exercise
   2.2.12) to deduce that a + b is a factor of ak + b” and write
   this polynomial as a product of two factors.


3. Sometimes a polynomial can be manipulated into a standard form for
   factorization. Write the following polynomials as difference of squares
   and thence factor them over Z. Determine their zeros.

(a) 4t2 - 20t - 13

(b) 5t2 - 6t + 1

(c) t4 - 47t2 + 1.

4. (a) Show that a manic cubic reducible over Z must have an integer
   zero.
   (b) Given that the cubic t3 - 8t2 + 33t - 42 is reducible over Z, factor
   it.


5. In factoring a polynomial, it is often useful to recall that the degree
   of the polynomial is the sum of the degrees of its factors. Prove that,
   if a polynomial of degree n is reducible, then it must have at least
   one factor whose degree does not exceed n/2.


6. One way to factor a polynomial is by the method of undelermined
   coefficients. A factorization of a certain form is assumed, and equa-
   tions satisfied by the coefficients of the factors is set up. Consider for
   example the problem of factoring the polynomial

t” + 98t4 + 1.