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------------------------------------------------ Factor the following expression completely.

[tex]\[ 6x^8 + 9x^7 + 33x^5 \][/tex]

To factor the expression [tex]$6x^8 + 9x^7 + 33x^5$[/tex] completely, follow these steps:

1. Identify the Greatest Common Factor (GCF):
- Look at all the terms: [tex]$6x^8$[/tex], [tex]$9x^7$[/tex], and [tex]$33x^5$[/tex].
- The coefficients are 6, 9, and 33. The greatest common factor of these numbers is 3.
- For the variable parts, pick the smallest power of [tex]$x$[/tex]: [tex]$x^5$[/tex].
- Therefore, the GCF is [tex]$3x^5$[/tex].

2. Factor out the GCF:
- Divide each term by the GCF [tex]$3x^5$[/tex]:
- [tex]$6x^8 \div 3x^5 = 2x^3$[/tex]
- [tex]$9x^7 \div 3x^5 = 3x^2$[/tex]
- [tex]$33x^5 \div 3x^5 = 11$[/tex]
- After factoring out, the expression inside the parentheses is [tex]$2x^3 + 3x^2 + 11$[/tex].

3. Write the factored expression:
- Combine the GCF with the simplified expression:
[tex]\[
6x^8 + 9x^7 + 33x^5 = 3x^5(2x^3 + 3x^2 + 11)
\][/tex]

This expression is now factored completely.