Questions: Factor the polynomial. If the polynomial cannot be factored, write prime. Factor out the greatest common factor as necessary. 3 s^2 + 14 s + 8 Select the correct choice below and fill in any answer boxes within your choice. A. 3 s^2 + 14 s + 8 = (Type your answer in factored form.) B. The polynomial is prime.

Factor the polynomial. If the polynomial cannot be factored, write prime. Factor out the greatest common factor as necessary.

3 s^2 + 14 s + 8

Select the correct choice below and fill in any answer boxes within your choice.
A. 3 s^2 + 14 s + 8 = (Type your answer in factored form.)
B. The polynomial is prime.

Transcript text: Factor the polynomial. If the polynomial cannot be factored, write prime. Factor out the greatest common factor as necessary. \[ 3 s^{2}+14 s+8 \] Select the correct choice below and fill in any answer boxes within your choice. A. $3 s^{2}+14 s+8=$ $\square$ (Type your answer in factored form.) B. The polynomial is prime.

Solution

Solution Steps

Step 1: Identify the Polynomial

We start with the polynomial: \[ 3s^2 + 14s + 8 \]

Step 2: Factor the Polynomial

To factor the polynomial, we look for two numbers that multiply to the product of the coefficient of the quadratic term (3) and the constant term (8), which is $3 \times 8 = 24$, and that also add up to the coefficient of the linear term (14).

The numbers that satisfy these conditions are 12 and 2 because: \[ 12 \times 2 = 24 \quad \text{and} \quad 12 + 2 = 14 \]

Step 3: Split the Middle Term

We split the middle term using the numbers found: \[ 3s^2 + 12s + 2s + 8 \]

Step 4: Factor by Grouping

Next, we group the terms and factor out the greatest common factor from each group: \[ 3s(s + 4) + 2(s + 4) \]

Step 5: Factor Out the Common Binomial Factor

We factor out the common binomial factor $(s + 4)$: \[ (3s + 2)(s + 4) \]