Question

select the correct answer.
using synthetic division, what is the factored form of this polynomial?
$3x^3 - 16x^2 + 12x + 16$
a. $(3x + 2)(x + 4)(x + 2)$
b. $(3x - 2)(x + 4)(x + 2)$
c. $(3x - 2)(x - 4)(x - 2)$
d. $(3x + 2)(x - 4)(x - 2)$

Explanation:

Step1: Find a root

We test possible rational roots using the Rational Root Theorem. The possible roots are factors of 16 over factors of 3: $\pm1, \pm2, \pm4, \pm8, \pm16, \pm\frac{1}{3}, \pm\frac{2}{3}, \pm\frac{4}{3}, \pm\frac{8}{3}, \pm\frac{16}{3}$. Let's test $x = 2$:
Substitute into $3x^3 - 16x^2 + 12x + 16$: $3(8) - 16(4) + 12(2) + 16 = 24 - 64 + 24 + 16 = 0$. So $x = 2$ is a root, and $(x - 2)$ is a factor.

Step2: Perform synthetic division

Using synthetic division with root 2:
Coefficients: 3 | -16 | 12 | 16
Bring down 3.
Multiply 3 by 2 = 6. Add to -16: -10.
Multiply -10 by 2 = -20. Add to 12: -8.
Multiply -8 by 2 = -16. Add to 16: 0.
The quotient polynomial is $3x^2 - 10x - 8$.

Step3: Factor the quadratic

Factor $3x^2 - 10x - 8$. We need two numbers that multiply to $3\times(-8) = -24$ and add to -10. Those numbers are -12 and 2.
Rewrite: $3x^2 - 12x + 2x - 8 = 3x(x - 4) + 2(x - 4) = (3x + 2)(x - 4)$.

Step4: Write the factored form

The original polynomial factors as $(x - 2)(3x + 2)(x - 4)$.

Answer:

D. $(3x + 2)(x - 4)(x - 2)$