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]
