Answer :
To factor the polynomial [tex]\(27x^{15} - 18x^{12} + 36x^9 + 45x^6\)[/tex] by the greatest common factor (GCF), we can follow these steps:
1. Identify the coefficients and variables:
- The terms in the polynomial are: [tex]\(27x^{15}\)[/tex], [tex]\(-18x^{12}\)[/tex], [tex]\(36x^9\)[/tex], and [tex]\(45x^6\)[/tex].
2. Find the GCF of the coefficients:
- The coefficients are 27, 18, 36, and 45.
- The greatest common factor of these numbers is 9.
3. Find the GCF of the variables:
- The exponents of [tex]\(x\)[/tex] in each term are 15, 12, 9, and 6.
- The greatest common factor of these exponents is 6, which corresponds to the smallest power of [tex]\(x\)[/tex].
4. Combine the GCF of the coefficients and variables:
- The overall greatest common factor of the entire polynomial is [tex]\(9x^6\)[/tex].
5. Factor out the GCF:
- Divide each term in the polynomial by the GCF [tex]\(9x^6\)[/tex]:
- [tex]\(27x^{15} \div 9x^6 = 3x^9\)[/tex]
- [tex]\(-18x^{12} \div 9x^6 = -2x^6\)[/tex]
- [tex]\(36x^9 \div 9x^6 = 4x^3\)[/tex]
- [tex]\(45x^6 \div 9x^6 = 5\)[/tex]
6. Write the factored expression:
- Factoring out [tex]\(9x^6\)[/tex], the polynomial becomes:
[tex]\[
9x^6 (3x^9 - 2x^6 + 4x^3 + 5)
\][/tex]
Therefore, the factored form of the polynomial [tex]\(27x^{15} - 18x^{12} + 36x^9 + 45x^6\)[/tex] by taking out the greatest common factor is:
[tex]\[ 9x^6 (3x^9 - 2x^6 + 4x^3 + 5) \][/tex]
1. Identify the coefficients and variables:
- The terms in the polynomial are: [tex]\(27x^{15}\)[/tex], [tex]\(-18x^{12}\)[/tex], [tex]\(36x^9\)[/tex], and [tex]\(45x^6\)[/tex].
2. Find the GCF of the coefficients:
- The coefficients are 27, 18, 36, and 45.
- The greatest common factor of these numbers is 9.
3. Find the GCF of the variables:
- The exponents of [tex]\(x\)[/tex] in each term are 15, 12, 9, and 6.
- The greatest common factor of these exponents is 6, which corresponds to the smallest power of [tex]\(x\)[/tex].
4. Combine the GCF of the coefficients and variables:
- The overall greatest common factor of the entire polynomial is [tex]\(9x^6\)[/tex].
5. Factor out the GCF:
- Divide each term in the polynomial by the GCF [tex]\(9x^6\)[/tex]:
- [tex]\(27x^{15} \div 9x^6 = 3x^9\)[/tex]
- [tex]\(-18x^{12} \div 9x^6 = -2x^6\)[/tex]
- [tex]\(36x^9 \div 9x^6 = 4x^3\)[/tex]
- [tex]\(45x^6 \div 9x^6 = 5\)[/tex]
6. Write the factored expression:
- Factoring out [tex]\(9x^6\)[/tex], the polynomial becomes:
[tex]\[
9x^6 (3x^9 - 2x^6 + 4x^3 + 5)
\][/tex]
Therefore, the factored form of the polynomial [tex]\(27x^{15} - 18x^{12} + 36x^9 + 45x^6\)[/tex] by taking out the greatest common factor is:
[tex]\[ 9x^6 (3x^9 - 2x^6 + 4x^3 + 5) \][/tex]
