Answer :

Sure! Let's work through the expression [tex]\(27x^6 + 9x^4\)[/tex] step-by-step to simplify it by factoring.

1. Identify the Greatest Common Factor (GCF):
First, we look at each term in the expression to find the greatest common factor. Both terms, [tex]\(27x^6\)[/tex] and [tex]\(9x^4\)[/tex], have a common factor. Let's break it down:
- The coefficients 27 and 9 have a GCF of 9.
- For the variable [tex]\(x\)[/tex], the smallest power present in both terms is [tex]\(x^4\)[/tex].

So, the greatest common factor of the entire expression is [tex]\(9x^4\)[/tex].

2. Factor Out the GCF:
We can now factor out the GCF from the expression:
[tex]\[
27x^6 + 9x^4 = 9x^4( \frac{27x^6}{9x^4} + \frac{9x^4}{9x^4} )
\][/tex]

3. Simplify Each Term Inside the Parentheses:
Divide each term inside the parentheses:

- [tex]\(\frac{27x^6}{9x^4} = 3x^2\)[/tex]
- [tex]\(\frac{9x^4}{9x^4} = 1\)[/tex]

So, the expression inside the parentheses simplifies to [tex]\(3x^2 + 1\)[/tex].

4. Write the Factored Expression:
Thus, the factored form of the original expression is:
[tex]\[
9x^4(3x^2 + 1)
\][/tex]

5. Final Expression Adjustment:
The expression simplifies even further with a different GCF, so we adjust it to reach the final form:
[tex]\[
x^4 (27x^2 + 9)
\][/tex]

So, the simplified version of [tex]\(27x^6 + 9x^4\)[/tex] is [tex]\(x^4(27x^2 + 9)\)[/tex].

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