We appreciate your visit to Factor the polynomial completely tex 6x 4 21x 3 15x 2 tex Select the correct choice below and if necessary fill in the answer box. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
Answer :
To factor the polynomial [tex]\(6x^4 + 21x^3 + 15x^2\)[/tex] completely, let's follow these steps:
1. Identify the Greatest Common Factor (GCF):
First, look for the greatest common factor of the coefficients and the common variable factor. The terms have coefficients 6, 21, and 15, which have a GCF of 3. Additionally, each term contains at least [tex]\(x^2\)[/tex] as a factor. Therefore, the GCF of the whole polynomial is [tex]\(3x^2\)[/tex].
2. Factor out the GCF:
Divide each term by the GCF, [tex]\(3x^2\)[/tex], and factor it out:
[tex]\[
6x^4 + 21x^3 + 15x^2 = 3x^2(2x^2 + 7x + 5)
\][/tex]
3. Factor the Remaining Quadratic Expression:
Now, focus on factoring the quadratic [tex]\(2x^2 + 7x + 5\)[/tex] completely. We want to find two numbers that multiply to [tex]\(2 \cdot 5 = 10\)[/tex] and add up to 7. These numbers are 2 and 5. Thus, we can rewrite:
[tex]\[
2x^2 + 7x + 5 = (x + 1)(2x + 5)
\][/tex]
4. Write the Completely Factored Form:
Substitute back into the expression:
[tex]\[
3x^2(2x^2 + 7x + 5) = 3x^2(x + 1)(2x + 5)
\][/tex]
So, the complete factorization of the polynomial [tex]\(6x^4 + 21x^3 + 15x^2\)[/tex] is:
[tex]\[
3x^2(x + 1)(2x + 5)
\][/tex]
Therefore, the correct choice is:
A. [tex]\(6x^4 + 21x^3 + 15x^2 = 3x^2(x + 1)(2x + 5)\)[/tex]
1. Identify the Greatest Common Factor (GCF):
First, look for the greatest common factor of the coefficients and the common variable factor. The terms have coefficients 6, 21, and 15, which have a GCF of 3. Additionally, each term contains at least [tex]\(x^2\)[/tex] as a factor. Therefore, the GCF of the whole polynomial is [tex]\(3x^2\)[/tex].
2. Factor out the GCF:
Divide each term by the GCF, [tex]\(3x^2\)[/tex], and factor it out:
[tex]\[
6x^4 + 21x^3 + 15x^2 = 3x^2(2x^2 + 7x + 5)
\][/tex]
3. Factor the Remaining Quadratic Expression:
Now, focus on factoring the quadratic [tex]\(2x^2 + 7x + 5\)[/tex] completely. We want to find two numbers that multiply to [tex]\(2 \cdot 5 = 10\)[/tex] and add up to 7. These numbers are 2 and 5. Thus, we can rewrite:
[tex]\[
2x^2 + 7x + 5 = (x + 1)(2x + 5)
\][/tex]
4. Write the Completely Factored Form:
Substitute back into the expression:
[tex]\[
3x^2(2x^2 + 7x + 5) = 3x^2(x + 1)(2x + 5)
\][/tex]
So, the complete factorization of the polynomial [tex]\(6x^4 + 21x^3 + 15x^2\)[/tex] is:
[tex]\[
3x^2(x + 1)(2x + 5)
\][/tex]
Therefore, the correct choice is:
A. [tex]\(6x^4 + 21x^3 + 15x^2 = 3x^2(x + 1)(2x + 5)\)[/tex]
Thanks for taking the time to read Factor the polynomial completely tex 6x 4 21x 3 15x 2 tex Select the correct choice below and if necessary fill in the answer box. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!
- Why do Businesses Exist Why does Starbucks Exist What Service does Starbucks Provide Really what is their product.
- The pattern of numbers below is an arithmetic sequence tex 14 24 34 44 54 ldots tex Which statement describes the recursive function used to..
- Morgan felt the need to streamline Edison Electric What changes did Morgan make.
Rewritten by : Barada
