College Algebra & Trigonometry, 2018a

2.8. ROOTS AND FACTORIZATION OF POLYNOMIALS 135
First we’ll graph the polynomial to see if we can find any real roots from the
graph:
40
20
x = −1.3
−4 −2 2 4
−20
−40
We can see in the graph that this polynomial has a root at x = − 4 . That means
3
that the polynomial must have a factor of 3x +4. We can use Synthetic Division
to find the other factor for this polynomial. Because we know that x = − 4 is a 3
root, we should get a zero remainder:
− 4 3
3 1 17 28
↓ −4 4 −28
3 −3 21 0
Notice that, because the root we used was a fraction, there is a common factor of
3 in the answer to our Synthetic Division. We should factor this out to obtain the
answer:
(x + 4 3 )(3x2 − 3x + 21) = (3x +4)(x 2 − x +7)
So, this means that:
3x 3 + x 2 +17x +28=0
(3x +4)(x 2 − x +7)=0
3x +4=0 x 2 − x +7=0
x = − 4 3
x ≈ 0.5 ± 2.598i