Questions: Factor. 3 s^6 - 192 s^3 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. 3 s^6 - 192 s^3 = (Factor completely.) B. The binomial is not factorable.

Factor.
3 s^6 - 192 s^3

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. 3 s^6 - 192 s^3 = (Factor completely.)
B. The binomial is not factorable.

Transcript text: Factor. \[ 3 s^{6}-192 s^{3} \] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $3 s^{6}-192 s^{3}=$ $\square$ (Factor completely.) B. The binomial is not factorable.

Solution

Solution Steps

To factor the expression $3s^6 - 192s^3$, we first look for a common factor in both terms. We notice that both terms have a factor of $3s^3$. After factoring out $3s^3$, we simplify the remaining expression. Finally, we check if the resulting expression can be factored further.

Step 1: Identify the Common Factor

The expression given is $3s^6 - 192s^3$. We first identify the greatest common factor (GCF) of the terms, which is $3s^3$.

Step 2: Factor Out the GCF

We factor out $3s^3$ from the expression: \[ 3s^6 - 192s^3 = 3s^3(s^3 - 64) \]

Step 3: Recognize and Factor the Difference of Cubes

The expression inside the parentheses, $s^3 - 64$, is a difference of cubes. We can use the formula for factoring a difference of cubes: \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \] Here, $a = s$ and $b = 4$, so: \[ s^3 - 64 = (s - 4)(s^2 + 4s + 16) \]

Step 4: Combine the Factors

Substituting back, we have: \[ 3s^6 - 192s^3 = 3s^3(s - 4)(s^2 + 4s + 16) \]