Answer :
Sure! Let's factor the Greatest Common Factor (GCF) out of the polynomial [tex]\(4x^5 + 20x^3 + 12x^2\)[/tex].
1. Identify the GCF of the coefficients:
- The coefficients in the polynomial are 4, 20, and 12.
- The GCF of 4, 20, and 12 is 4.
2. Identify the GCF of the variable parts:
- The variable parts are [tex]\(x^5\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^2\)[/tex].
- The smallest exponent of [tex]\(x\)[/tex] in the terms is 2. Therefore, the GCF of the variable part is [tex]\(x^2\)[/tex].
3. Combine the GCFs:
- The overall GCF of the polynomial is [tex]\(4x^2\)[/tex].
4. Factor out the GCF from each term in the polynomial:
- [tex]\(4x^5 \div 4x^2 = x^3\)[/tex]
- [tex]\(20x^3 \div 4x^2 = 5x\)[/tex]
- [tex]\(12x^2 \div 4x^2 = 3\)[/tex]
5. Rewrite the polynomial as the product of the GCF and the remaining polynomial:
- So, [tex]\(4x^5 + 20x^3 + 12x^2\)[/tex] factored out becomes [tex]\(4x^2(x^3 + 5x + 3)\)[/tex].
Therefore, the polynomial [tex]\(4x^5 + 20x^3 + 12x^2\)[/tex] factored by the GCF is:
[tex]\[ 4x^2(x^3 + 5x + 3) \][/tex]
1. Identify the GCF of the coefficients:
- The coefficients in the polynomial are 4, 20, and 12.
- The GCF of 4, 20, and 12 is 4.
2. Identify the GCF of the variable parts:
- The variable parts are [tex]\(x^5\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^2\)[/tex].
- The smallest exponent of [tex]\(x\)[/tex] in the terms is 2. Therefore, the GCF of the variable part is [tex]\(x^2\)[/tex].
3. Combine the GCFs:
- The overall GCF of the polynomial is [tex]\(4x^2\)[/tex].
4. Factor out the GCF from each term in the polynomial:
- [tex]\(4x^5 \div 4x^2 = x^3\)[/tex]
- [tex]\(20x^3 \div 4x^2 = 5x\)[/tex]
- [tex]\(12x^2 \div 4x^2 = 3\)[/tex]
5. Rewrite the polynomial as the product of the GCF and the remaining polynomial:
- So, [tex]\(4x^5 + 20x^3 + 12x^2\)[/tex] factored out becomes [tex]\(4x^2(x^3 + 5x + 3)\)[/tex].
Therefore, the polynomial [tex]\(4x^5 + 20x^3 + 12x^2\)[/tex] factored by the GCF is:
[tex]\[ 4x^2(x^3 + 5x + 3) \][/tex]
