# Algebra: operations with polynomials

Section 5.1

Greatest Common Factor for Monomials

Find the greatest common factor for each group of monomials.

40. 16x²z, 40xz², 72x³

Section 5.2

Factoring a Difference of Two Squares

Factor each polynomial.

16. 9a² - 64b²

Section 5.3

Factoring with Two Variables

Factor each polynomial.

64. h²- 9hs + 9s²

Section 5.4

The ac Method

Factor each trinomial using the ac method.

26. 21x² + 2x - 3

Section 5.5

Factoring a Difference of Two Fourth Powers

Factor each polynomial completely.

36. m⁴ - n⁴

Section 5.6

The Zero Factor Property

Solve by factoring.

32. 2w(4w + 1) = 1

Discussion Questions

DQ 1: How would you use this procedure to determine if a given polynomial is prime, that is, determine that it cannot be factored into a product of two linear polynomials (note pps 321 and 327 of the text, Prime Polynomials)?

(7) P(x) = ax2 + bx + c, where a ≠ 1,

DQ 2: Given the prime factorization of an integer, how can you determine if our integer is a perfect square?

DQ 3: In the procedure for determining the prime factorization of an integer, why is it that we need not consider dividing by prime factors greater than the square root of that integer?

DQ 4: In the prime factorization of an integer, what is the maximum number of prime factors greater than the square root of that integer? If there is a prime factor greater than the square root of that integer, can the integer be a perfect square?

Note the polynomial form P(x) = x2 - a. And, note that the two linear factors ( x + sqrt(a) ) ( x - sqrt(a) ), when multiplied together, yield

( x + sqrt(a) ) ( x - sqrt(a) ) = x2 - a.

Consequently, we note that an equation of the form

x2 - a = 0

can be factored as

( x + sqrt(a) ) ( x - sqrt(a) ) = 0,

giving the two zeros of our equation

x = sqrt(a), -sqrt(a).

Team Exercise Two:

Consider the quadratic expression representing displacement of an object thrown toward earth at an initial velocity of 32 ft / sec from a height of 128 ft.

h(t) = -16t2 - 32t + 128

For this expression, determine the GCF (Greatest Common Factor) of our three coefficients and, as in exercise one, express h(t) as a product of this GCF and a resulting trinomial. Factor our resulting trinomial into a product of two binomials, thus completely factoring our expression h(t). What does a solution, or zero, of the equation

-16t2 - 32t + 128 = 0

represent? Determine the two solutions.

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#### Solution Summary

This solution performs many operations in factoring polynomials.

Algebra Operations with Polynomials

1. For each polynomial listed below, determine

i the degree of the polynomial

ii the coefficient of the leading term

iii the constant term

a. P(x) = x + 1

b. Q(x) = 3x + 2

c. R(x) = x2 + 2x + 1

d. W(x) = 4x2 + x + 3

e. Z(x) = 3x3 + 2x2 + x

Questions 2 through 8 below refer to the polynomials P(x), Q(x), R(x), and Z(x) of question #1.

2. What is the value of P(1), that is, P(x) above evaluated at x = 1?

3. What is the value of Z(1), that is, Z(x) evaluated at x = 1?

4. What is the value of Z(2)?

5. What is the polynomial resulting from the polynomial addition P(x) + Q(x)?

6. What is the polynomial resulting from the polynomial subtraction Q(x) - P(x)?

7. What is the polynomial resulting from the polynomial multiplication P(x) * Q(x)?

8. What is the polynomial resulting from the polynomial division R(x) / P(x)?

9. Simplify

a. (5x4)3

b. (-3x2)3

10. Simplify. Write answers without negative exponents

a. (-2x-2y)-3

b. (-3x-3y)2