? Question

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. 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]