Answer :
To factor the greatest common factor (GCF) out of the polynomial [tex]\(20x^6 + 12x^4 + 8x^3\)[/tex], follow these steps:
1. Identify the GCF of the coefficients:
- The coefficients of the polynomial are 20, 12, and 8.
- The GCF of 20, 12, and 8 is 4. This is the largest number that divides all of them evenly.
2. Identify the common variable factor:
- The terms in the polynomial are [tex]\(x^6\)[/tex], [tex]\(x^4\)[/tex], and [tex]\(x^3\)[/tex].
- The lowest power of [tex]\(x\)[/tex] common to all terms is [tex]\(x^3\)[/tex].
3. Combine the GCF of the coefficients and the common variable factor:
- The overall GCF of the polynomial is [tex]\(4x^3\)[/tex].
4. Factor the GCF out of the polynomial:
- Divide each term of the polynomial by the GCF [tex]\(4x^3\)[/tex]:
- [tex]\(20x^6 \div 4x^3 = 5x^3\)[/tex]
- [tex]\(12x^4 \div 4x^3 = 3x\)[/tex]
- [tex]\(8x^3 \div 4x^3 = 2\)[/tex]
5. Write the factored form:
- The polynomial can be rewritten by factoring out the GCF:
[tex]\[
20x^6 + 12x^4 + 8x^3 = 4x^3(5x^3 + 3x + 2)
\][/tex]
In this solution, the GCF is [tex]\(4x^3\)[/tex] and the remaining polynomial after factoring out the GCF is [tex]\(5x^3 + 3x + 2\)[/tex].
1. Identify the GCF of the coefficients:
- The coefficients of the polynomial are 20, 12, and 8.
- The GCF of 20, 12, and 8 is 4. This is the largest number that divides all of them evenly.
2. Identify the common variable factor:
- The terms in the polynomial are [tex]\(x^6\)[/tex], [tex]\(x^4\)[/tex], and [tex]\(x^3\)[/tex].
- The lowest power of [tex]\(x\)[/tex] common to all terms is [tex]\(x^3\)[/tex].
3. Combine the GCF of the coefficients and the common variable factor:
- The overall GCF of the polynomial is [tex]\(4x^3\)[/tex].
4. Factor the GCF out of the polynomial:
- Divide each term of the polynomial by the GCF [tex]\(4x^3\)[/tex]:
- [tex]\(20x^6 \div 4x^3 = 5x^3\)[/tex]
- [tex]\(12x^4 \div 4x^3 = 3x\)[/tex]
- [tex]\(8x^3 \div 4x^3 = 2\)[/tex]
5. Write the factored form:
- The polynomial can be rewritten by factoring out the GCF:
[tex]\[
20x^6 + 12x^4 + 8x^3 = 4x^3(5x^3 + 3x + 2)
\][/tex]
In this solution, the GCF is [tex]\(4x^3\)[/tex] and the remaining polynomial after factoring out the GCF is [tex]\(5x^3 + 3x + 2\)[/tex].
