Answer :
To factor the expression [tex]\(12x^2 + 28x + 8\)[/tex], we need to follow these steps:
1. Find the Greatest Common Factor (GCF):
First, identify the GCF of all the coefficients in the polynomial. The coefficients are 12, 28, and 8.
- The GCF of 12, 28, and 8 is 4.
2. Factor Out the GCF:
Factor out the GCF, which is 4, from the entire expression:
[tex]\[
12x^2 + 28x + 8 = 4(3x^2 + 7x + 2)
\][/tex]
3. Factor the Quadratic Inside the Parentheses:
Now, let's focus on factoring the quadratic expression inside the parentheses: [tex]\(3x^2 + 7x + 2\)[/tex].
Look for two numbers that multiply to the product of the leading coefficient (3) and the constant term (2), which is [tex]\(3 \times 2 = 6\)[/tex], and add up to the middle coefficient (7).
- The numbers that satisfy this are 6 and 1, because:
- [tex]\(6 \times 1 = 6\)[/tex]
- [tex]\(6 + 1 = 7\)[/tex]
4. Rewrite and Factor by Grouping:
Rewrite the middle term using the numbers found:
[tex]\[
3x^2 + 7x + 2 = 3x^2 + 6x + 1x + 2
\][/tex]
Group the terms:
[tex]\[
= (3x^2 + 6x) + (1x + 2)
\][/tex]
Factor each group:
- Factor out the GCF from the first group: [tex]\(3x(x + 2)\)[/tex]
- Factor out the GCF from the second group: [tex]\(1(x + 2)\)[/tex] or simply [tex]\((x + 2)\)[/tex]
Combine the groups:
[tex]\[
= (3x + 1)(x + 2)
\][/tex]
5. Put It All Together:
Combine the factored terms with the GCF at the beginning:
[tex]\[
4(3x^2 + 7x + 2) = 4(3x + 1)(x + 2)
\][/tex]
So, the fully factored form of the expression [tex]\(12x^2 + 28x + 8\)[/tex] is [tex]\(4(x + 2)(3x + 1)\)[/tex].
1. Find the Greatest Common Factor (GCF):
First, identify the GCF of all the coefficients in the polynomial. The coefficients are 12, 28, and 8.
- The GCF of 12, 28, and 8 is 4.
2. Factor Out the GCF:
Factor out the GCF, which is 4, from the entire expression:
[tex]\[
12x^2 + 28x + 8 = 4(3x^2 + 7x + 2)
\][/tex]
3. Factor the Quadratic Inside the Parentheses:
Now, let's focus on factoring the quadratic expression inside the parentheses: [tex]\(3x^2 + 7x + 2\)[/tex].
Look for two numbers that multiply to the product of the leading coefficient (3) and the constant term (2), which is [tex]\(3 \times 2 = 6\)[/tex], and add up to the middle coefficient (7).
- The numbers that satisfy this are 6 and 1, because:
- [tex]\(6 \times 1 = 6\)[/tex]
- [tex]\(6 + 1 = 7\)[/tex]
4. Rewrite and Factor by Grouping:
Rewrite the middle term using the numbers found:
[tex]\[
3x^2 + 7x + 2 = 3x^2 + 6x + 1x + 2
\][/tex]
Group the terms:
[tex]\[
= (3x^2 + 6x) + (1x + 2)
\][/tex]
Factor each group:
- Factor out the GCF from the first group: [tex]\(3x(x + 2)\)[/tex]
- Factor out the GCF from the second group: [tex]\(1(x + 2)\)[/tex] or simply [tex]\((x + 2)\)[/tex]
Combine the groups:
[tex]\[
= (3x + 1)(x + 2)
\][/tex]
5. Put It All Together:
Combine the factored terms with the GCF at the beginning:
[tex]\[
4(3x^2 + 7x + 2) = 4(3x + 1)(x + 2)
\][/tex]
So, the fully factored form of the expression [tex]\(12x^2 + 28x + 8\)[/tex] is [tex]\(4(x + 2)(3x + 1)\)[/tex].
