Answer :
To factor the greatest common factor (GCF) out of the polynomial [tex]\(12x^5 + 8x^3 + 28x^2\)[/tex], follow these steps:
1. Identify the coefficients of the polynomial:
- The polynomial is composed of terms with coefficients 12, 8, and 28.
2. Find the GCF of the coefficients:
- List the factors for each coefficient:
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 8: 1, 2, 4, 8
- Factors of 28: 1, 2, 4, 7, 14, 28
- The greatest common factor among 12, 8, and 28 is 4.
3. Identify the smallest power of [tex]\(x\)[/tex] common to all terms:
- Look at the terms [tex]\(x^5\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^2\)[/tex].
- The smallest power of [tex]\(x\)[/tex] common to all terms is [tex]\(x^2\)[/tex].
4. Factor out the GCF and the smallest power of [tex]\(x\)[/tex]:
- The GCF of the numerical coefficients is 4 and the smallest power of [tex]\(x\)[/tex] is [tex]\(x^2\)[/tex].
- Factor [tex]\(4x^2\)[/tex] out of each term in the polynomial:
[tex]\[
12x^5 + 8x^3 + 28x^2 = 4x^2(3x^3) + 4x^2(2x) + 4x^2(7)
\][/tex]
5. Write the factored form of the polynomial:
- After factoring out [tex]\(4x^2\)[/tex], the expression inside the parentheses is the remaining part of each term after division:
[tex]\[
4x^2(3x^3 + 2x + 7)
\][/tex]
Therefore, the polynomial [tex]\(12x^5 + 8x^3 + 28x^2\)[/tex] factors to [tex]\(4x^2(3x^3 + 2x + 7)\)[/tex].
1. Identify the coefficients of the polynomial:
- The polynomial is composed of terms with coefficients 12, 8, and 28.
2. Find the GCF of the coefficients:
- List the factors for each coefficient:
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 8: 1, 2, 4, 8
- Factors of 28: 1, 2, 4, 7, 14, 28
- The greatest common factor among 12, 8, and 28 is 4.
3. Identify the smallest power of [tex]\(x\)[/tex] common to all terms:
- Look at the terms [tex]\(x^5\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^2\)[/tex].
- The smallest power of [tex]\(x\)[/tex] common to all terms is [tex]\(x^2\)[/tex].
4. Factor out the GCF and the smallest power of [tex]\(x\)[/tex]:
- The GCF of the numerical coefficients is 4 and the smallest power of [tex]\(x\)[/tex] is [tex]\(x^2\)[/tex].
- Factor [tex]\(4x^2\)[/tex] out of each term in the polynomial:
[tex]\[
12x^5 + 8x^3 + 28x^2 = 4x^2(3x^3) + 4x^2(2x) + 4x^2(7)
\][/tex]
5. Write the factored form of the polynomial:
- After factoring out [tex]\(4x^2\)[/tex], the expression inside the parentheses is the remaining part of each term after division:
[tex]\[
4x^2(3x^3 + 2x + 7)
\][/tex]
Therefore, the polynomial [tex]\(12x^5 + 8x^3 + 28x^2\)[/tex] factors to [tex]\(4x^2(3x^3 + 2x + 7)\)[/tex].
