College Algebra & Trigonometry, 2018a

2.8. ROOTS AND FACTORIZATION OF POLYNOMIALS 133
So, now we know that 2x 3 − 3x 2 +2x − 8=(x − 2)(2x 2 + x +4). To finish the
problem, we can set each factor equal to zero and find the roots:
2x 3 − 3 2 +2x − 8=0
(x − 2)(2x 2 + x +4)=0
x − 2=0 2x 2 + x +4=0
x =2 x ≈−0.25 ± 1.392i
Let’s look at an example that has more than one real root:
Example
Find all real and complex roots for the given equation. Express the given polynomial
as the product of prime factors with integer coefficients.
3x 4 +5x 3 − 45x 2 +19x − 30 = 0
First we’ll graph the polynomial to see if we can find any real roots from the
graph:
400
200
x = −5 x =3
−8 −6 −4 −2 2 4 6 8
−200
−400