College Algebra & Trigonometry, 2018a

132 CHAPTER 2. POLYNOMIAL AND RATIONAL FUNCTIONS
2.8 Roots and Factorization of Polynomials
In this section we will use some of the skills we have seen in previous sections
in order to find all the roots of a polynomial function (both real and complex)
and also factor the polynomial as the product of prime factors with integer coefficients.
Example
Find all real and complex roots for the given equation. Express the given polynomial
as the product of prime factors with integer coefficients.
2x 3 − 3x 2 +2x − 8=0
First we’ll graph the polynomial to see if we can find any real roots from the
graph:
20
10
x =2
−4 −2 2 4
−10
−20
We can see that there is a root at x =2. This means that the polynomial will
have a factor of (x − 2). We can use Synthetic Division to find any other factors.
Because x =2is a root, we should get a zero remainder:
2 2 −3 2 −8
↓ 4 2 8
2 1 4 0