Join the conversation on IDNLearn.com and get the answers you seek from experts. Get prompt and accurate answers to your questions from our community of knowledgeable experts.

Use the grouping method to factor this polynomial completely.

[tex]\[3x^3 + 12x^2 + 2x + 8\][/tex]

A. [tex]\(\left(3x^2 + 4\right)(x + 4)\)[/tex]

B. [tex]\(\left(3x^2 + 4\right)(x + 2)\)[/tex]

C. [tex]\(\left(3x^2 + 2\right)(x + 4)\)[/tex]

D. [tex]\(\left(3x^2 + 2\right)(x + 2)\)[/tex]


Sagot :

To factor the polynomial [tex]\(3x^3 + 12x^2 + 2x + 8\)[/tex] completely using the grouping method, follow these steps:

1. Group the terms: We start by grouping the terms in pairs to make it easier to factor by grouping.
[tex]\[ 3x^3 + 12x^2 + 2x + 8 = (3x^3 + 12x^2) + (2x + 8) \][/tex]

2. Factor out the greatest common factor (GCF) from each pair of terms:
- For the first group, [tex]\(3x^3 + 12x^2\)[/tex], the GCF is [tex]\(3x^2\)[/tex]. So, we factor [tex]\(3x^2\)[/tex] out:
[tex]\[ 3x^3 + 12x^2 = 3x^2(x + 4) \][/tex]
- For the second group, [tex]\(2x + 8\)[/tex], the GCF is [tex]\(2\)[/tex]. So, we factor [tex]\(2\)[/tex] out:
[tex]\[ 2x + 8 = 2(x + 4) \][/tex]

Now our expression looks like this:
[tex]\[ 3x^2(x + 4) + 2(x + 4) \][/tex]

3. Factor out the common binomial factor: Notice that both terms now contain the common binomial factor [tex]\((x + 4)\)[/tex]. We factor [tex]\((x + 4)\)[/tex] out:
[tex]\[ 3x^2(x + 4) + 2(x + 4) = (x + 4)(3x^2 + 2) \][/tex]

So, the polynomial [tex]\(3x^3 + 12x^2 + 2x + 8\)[/tex] factors completely to [tex]\((x + 4)(3x^2 + 2)\)[/tex].

Therefore, the correct answer is:
[tex]\[ \boxed{C. (3x^2 + 2)(x + 4)} \][/tex]
Thank you for being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. IDNLearn.com has the solutions to your questions. Thanks for stopping by, and come back for more insightful information.