To factor the polynomial expression [tex]\(3x^3 + 3x^2 - 18x\)[/tex], we can follow these steps:
1. Factor out the greatest common factor (GCF):
First, observe that each term in the polynomial shares a common factor of [tex]\(3x\)[/tex]. We can factor out [tex]\(3x\)[/tex] from the expression:
[tex]\[
3x (x^2 + x - 6)
\][/tex]
2. Factor the quadratic expression inside the parentheses:
Now, we need to factor the quadratic expression [tex]\(x^2 + x - 6\)[/tex]. To do this, we look for two numbers that multiply to [tex]\(-6\)[/tex] (the constant term) and add up to [tex]\(1\)[/tex] (the coefficient of the linear term).
The numbers [tex]\(-2\)[/tex] and [tex]\(3\)[/tex] meet these criteria because:
[tex]\[
-2 \cdot 3 = -6
\][/tex]
[tex]\[
-2 + 3 = 1
\][/tex]
Therefore, we can write the quadratic expression as:
[tex]\[
x^2 + x - 6 = (x - 2)(x + 3)
\][/tex]
3. Combine the factored terms:
Substitute the factored quadratic back into the expression:
[tex]\[
3x (x - 2)(x + 3)
\][/tex]
Therefore, the fully factored form of the polynomial [tex]\(3x^3 + 3x^2 - 18x\)[/tex] is:
[tex]\[
3x (x - 2)(x + 3)
\][/tex]
Among the given options, the correct answer is:
[tex]\[
3x (x + 3)(x - 2)
\][/tex]
