What is the completely factored form of this polynomial?

[tex]\[ x^3 + 3x^2 - 6x - 18 \][/tex]

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

B. [tex]\(\left(x^2 - 6\right)(x + 3)\)[/tex]

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

D. [tex]\((x + 6)(x - 1)(x + 3)\)[/tex]



Answer :

To find the completely factored form of the polynomial [tex]\( x^3 + 3x^2 - 6x - 18 \)[/tex], we can follow a structured approach. Here is a detailed, step-by-step solution:

1. Identify the polynomial:
We are working with the polynomial:
[tex]\[ x^3 + 3x^2 - 6x - 18 \][/tex]

2. Factor the polynomial:
To factor the given polynomial, we attempt to express it as a product of simpler polynomials.

3. Consider factoring by grouping (if applicable):
Let’s split the polynomial into two groups:
[tex]\[ (x^3 + 3x^2) + (-6x - 18) \][/tex]

Factor out the common factors in each group:
[tex]\[ x^2(x + 3) - 6(x + 3) \][/tex]

Notice that [tex]\((x + 3)\)[/tex] is a common factor:
[tex]\[ (x^2 - 6)(x + 3) \][/tex]

4. Write down the factored form:
Thus, the polynomial [tex]\( x^3 + 3x^2 - 6x - 18 \)[/tex] can be factored as:
[tex]\[ (x^2 - 6)(x + 3) \][/tex]

5. Verify the factorization by expanding the factored form:
[tex]\[ (x + 3)(x^2 - 6) = x^3 + 3x^2 - 6x - 18 \][/tex]

This matches the original polynomial, confirming our factorization is correct.

6. Select the correct answer:
Based on the factorization, we see that the completely factored form of the polynomial is:

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

Therefore, the correct answer is:
[tex]\[ \boxed{B. (x^2 - 6)(x + 3)} \][/tex]