Find answers to your questions faster and easier with IDNLearn.com. Ask anything and receive well-informed answers from our community of experienced professionals.
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]
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 using this platform to share and learn. Keep asking and answering. We appreciate every contribution you make. Thank you for choosing IDNLearn.com for your queries. We’re here to provide accurate answers, so visit us again soon.