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Answer :
To factor out the greatest common factor from the polynomial [tex]\(9x^7 - 72x^6 + 27x^5\)[/tex], we should follow these steps:
1. Identify the Greatest Common Factor (GCF) of the coefficients:
- Look at the coefficients of each term in the polynomial: 9, -72, and 27.
- The GCF of these numbers is 9.
2. Determine the common factor for the variable part:
- Notice the variable [tex]\(x\)[/tex] appears in all terms, with exponents 7, 6, and 5.
- The smallest exponent is 5, so the GCF for the variable part is [tex]\(x^5\)[/tex].
3. Combine these to get the overall GCF:
- These observations lead us to conclude that the greatest common factor of the entire polynomial is [tex]\(9x^5\)[/tex].
4. Factor out the GCF from each term:
- Divide each term of the polynomial by [tex]\(9x^5\)[/tex]:
[tex]\[
\frac{9x^7}{9x^5} = x^2
\][/tex]
[tex]\[
\frac{-72x^6}{9x^5} = -8x
\][/tex]
[tex]\[
\frac{27x^5}{9x^5} = 3
\][/tex]
5. Write the polynomial in its factored form:
- After factoring out [tex]\(9x^5\)[/tex], the polynomial can be written as:
[tex]\[
9x^7 - 72x^6 + 27x^5 = 9x^5(x^2 - 8x + 3)
\][/tex]
Therefore, the factored form of the polynomial is:
[tex]\[ 9x^5(x^2 - 8x + 3) \][/tex]
This is the correct factorization with the greatest common factor factored out.
1. Identify the Greatest Common Factor (GCF) of the coefficients:
- Look at the coefficients of each term in the polynomial: 9, -72, and 27.
- The GCF of these numbers is 9.
2. Determine the common factor for the variable part:
- Notice the variable [tex]\(x\)[/tex] appears in all terms, with exponents 7, 6, and 5.
- The smallest exponent is 5, so the GCF for the variable part is [tex]\(x^5\)[/tex].
3. Combine these to get the overall GCF:
- These observations lead us to conclude that the greatest common factor of the entire polynomial is [tex]\(9x^5\)[/tex].
4. Factor out the GCF from each term:
- Divide each term of the polynomial by [tex]\(9x^5\)[/tex]:
[tex]\[
\frac{9x^7}{9x^5} = x^2
\][/tex]
[tex]\[
\frac{-72x^6}{9x^5} = -8x
\][/tex]
[tex]\[
\frac{27x^5}{9x^5} = 3
\][/tex]
5. Write the polynomial in its factored form:
- After factoring out [tex]\(9x^5\)[/tex], the polynomial can be written as:
[tex]\[
9x^7 - 72x^6 + 27x^5 = 9x^5(x^2 - 8x + 3)
\][/tex]
Therefore, the factored form of the polynomial is:
[tex]\[ 9x^5(x^2 - 8x + 3) \][/tex]
This is the correct factorization with the greatest common factor factored out.
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