Question

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

Explanation:

Step1: Find a root

Try possible rational roots using Rational Root Theorem. Possible roots are $\pm1, \pm2, \pm4, \pm8, \pm16, \pm\frac{1}{3}, \pm\frac{2}{3}, \pm\frac{4}{3}, \pm\frac{8}{3}, \pm\frac{16}{3}$. Test $x = 2$: $3(2)^3 - 16(2)^2 + 12(2) + 16 = 24 - 64 + 24 + 16 = -0$. So $x = 2$ is a root, so $(x - 2)$ is a factor.

Step2: Perform synthetic division

Divide $3x^3 - 16x^2 + 12x + 16$ by $(x - 2)$ using synthetic division:
Coefficients: 3 | -16 | 12 | 16
Bring down 3. Multiply by 2: 6. Add to -16: -10. Multiply by 2: -20. Add to 12: -8. Multiply by 2: -16. Add to 16: 0. So quotient is $3x^2 - 10x - 8$.

Step3: Factor quadratic

Factor $3x^2 - 10x - 8$. Find two numbers: $-12$ and $2$ (since $-12 \times 2 = -24$ and $-12 + 2 = -10$). Rewrite: $3x^2 - 12x + 2x - 8 = 3x(x - 4) + 2(x - 4) = (3x + 2)(x - 4)$.

Step4: Combine factors

So the polynomial factors as $(x - 2)(3x + 2)(x - 4)$, which is $(3x + 2)(x - 4)(x - 2)$.

Answer:

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