Answer :

Let's factor the expression [tex]\(9x^9 - 63x^7\)[/tex] step-by-step:

1. Identify the Greatest Common Factor (GCF):
The expression is [tex]\(9x^9 - 63x^7\)[/tex]. First, we need to identify any common factors in the coefficients and the powers of [tex]\(x\)[/tex].

- The coefficients are 9 and 63. The greatest common factor of 9 and 63 is 9.
- Both terms have [tex]\(x\)[/tex] in them, with powers of 9 and 7. The smallest power is 7, so the GCF for the variable part is [tex]\(x^7\)[/tex].

Therefore, the GCF of the entire expression is [tex]\(9x^7\)[/tex].

2. Factor out the GCF:
Once we have identified the GCF, we factor it out from each term in the expression.

- The original expression is [tex]\(9x^9 - 63x^7\)[/tex].
- Factor out [tex]\(9x^7\)[/tex]:

[tex]\[
9x^9 - 63x^7 = 9x^7(x^2) - 9x^7(7)
\][/tex]

Simplifying each term inside the parentheses, we have:

[tex]\[
9x^7(x^2 - 7)
\][/tex]

3. Check the Factorization:
To ensure the factorization is correct, you can expand the expression back out to see if it matches the original:

- Expand [tex]\(9x^7(x^2 - 7)\)[/tex]:

[tex]\[
9x^7 \times x^2 = 9x^9
\][/tex]
[tex]\[
9x^7 \times (-7) = -63x^7
\][/tex]

- The expanded form [tex]\(9x^9 - 63x^7\)[/tex] matches the original expression.

Thus, the completely factored form of the expression is:

[tex]\[
9x^7(x^2 - 7)
\][/tex]

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