Use the grouping method to factor [tex]2x^3 + 6x^2 - 7x - 21[/tex].

A. [tex](x-3)(x+7)[/tex]
B. [tex](x+3)(2x^2 - 7)[/tex]
C. [tex]2x(x+3)(x-7)[/tex]
D. [tex](x-3)(2x^2 + 7)[/tex]



Answer :

Certainly! Let's use the grouping method to factor the polynomial [tex]\(2x^3 + 6x^2 - 7x - 21\)[/tex]. Here’s a step-by-step solution:

1. Rewrite the polynomial in two groups:
[tex]\[2x^3 + 6x^2 - 7x - 21\][/tex]

Let's divide this polynomial into two parts:
[tex]\[ (2x^3 + 6x^2) + (-7x - 21) \][/tex]

2. Factor out the greatest common factor (GCF) in each group:
- From the first group [tex]\(2x^3 + 6x^2\)[/tex], the GCF is [tex]\(2x^2\)[/tex]:
[tex]\[ 2x^2(x + 3) \][/tex]

- From the second group [tex]\(-7x - 21\)[/tex], the GCF is [tex]\(-7\)[/tex]:
[tex]\[ -7(x + 3) \][/tex]

Now the polynomial looks like:
[tex]\[ 2x^2(x + 3) - 7(x + 3) \][/tex]

3. Factor out the common binomial factor:
We see that [tex]\((x + 3)\)[/tex] is a common factor in both terms:
[tex]\[ (x + 3)(2x^2 - 7) \][/tex]

Therefore, the factored form of the polynomial [tex]\(2x^3 + 6x^2 - 7x - 21\)[/tex] is:
[tex]\[ (x + 3)(2x^2 - 7) \][/tex]

4. Compare with the given options:
We can observe that this matches option B:
[tex]\[ (x + 3)(2x^2 - 7) \][/tex]

So, the correct answer is:
[tex]\[ \boxed{(x + 3)\left(2x^2 - 7\right)} \][/tex]

Thus, the correct option is [tex]\( \text{B} \)[/tex].