Factor this polynomial expression, and then write it in its fully factored form.

[tex]\[ 3x^3 + 3x^2 - 18x \][/tex]

Select the correct answer.

A. [tex]\(3x(x-3)(x+2)\)[/tex]

B. [tex]\((3x^2 + 9x)(x-2)\)[/tex]

C. [tex]\(3x(x^2 + x - 6)\)[/tex]

D. [tex]\(3x(x+3)(x-2)\)[/tex]



Answer :

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]