To factor the polynomial expression [tex]\(3x^3 + 3x^2 - 18x\)[/tex], we can follow these steps:
1. Identify the greatest common factor (GCF):
Look for a common factor in all the terms of the polynomial. The terms [tex]\(3x^3\)[/tex], [tex]\(3x^2\)[/tex], and [tex]\(-18x\)[/tex] all have a common factor of [tex]\(3x\)[/tex].
2. Factor out the GCF:
We factor [tex]\(3x\)[/tex] out from each term:
[tex]\[
3x^3 + 3x^2 - 18x = 3x(x^2 + x - 6)
\][/tex]
3. Factor the quadratic expression:
Next, we need to factor the quadratic expression inside the parentheses, [tex]\(x^2 + x - 6\)[/tex]. 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 [tex]\(x\)[/tex] term).
These two numbers are [tex]\(3\)[/tex] and [tex]\(-2\)[/tex], since [tex]\(3 \cdot (-2) = -6\)[/tex] and [tex]\(3 + (-2) = 1\)[/tex].
4. Express the quadratic as a product of binomials:
Rewrite the quadratic expression using these two numbers:
[tex]\[
x^2 + x - 6 = (x + 3)(x - 2)
\][/tex]
5. Combine the results:
Substitute the factored quadratic expression back into the overall expression:
[tex]\[
3x(x^2 + x - 6) = 3x(x + 3)(x - 2)
\][/tex]
Thus, the fully factored form of the polynomial [tex]\(3x^3 + 3x^2 - 18x\)[/tex] is:
[tex]\[
3x(x + 3)(x - 2)
\][/tex]
So, the correct answer is:
[tex]\[
3x(x + 3)(x - 2)
\][/tex]
