Answer :

Sure! Let's go through the process of factoring the polynomial step-by-step:

The polynomial given is:
[tex]\[ 9x^7 - 72x^6 + 72x^5 \][/tex]

Step 1: Identify the Greatest Common Factor (GCF):

First, we need to find the greatest common factor (GCF) of the coefficients and the variables involved in each term.

- The coefficients are 9, 72, and 72. The GCF of these numbers is 9.
- The variable [tex]\(x\)[/tex] is present in each term, and the lowest power of [tex]\(x\)[/tex] is [tex]\(x^5\)[/tex].

Therefore, the GCF of the polynomial is [tex]\(9x^5\)[/tex].

Step 2: Factor out the GCF:

We divide each term of the polynomial by the GCF [tex]\(9x^5\)[/tex]:

[tex]\[
9x^7 \div 9x^5 = x^2
\][/tex]

[tex]\[
-72x^6 \div 9x^5 = -8x
\][/tex]

[tex]\[
72x^5 \div 9x^5 = 8
\][/tex]

Putting it together, the polynomial factors as:
[tex]\[
9x^5(x^2 - 8x + 8)
\][/tex]

So, the factored form of the polynomial [tex]\(9x^7 - 72x^6 + 72x^5\)[/tex] is:
[tex]\[
9x^5(x^2 - 8x + 8)
\][/tex]

I hope this breakdown helps you understand how to factor the polynomial! If you have any questions, feel free to ask.

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