Question: Factor out the common factor. (Factor your answer completely.)
$$12x^5y^4 - 9xy^5 + 27x^2y^8$$
Answer given: $$3xy^4(4x^4y - 3y + 9xy^4)$$
1. **State the problem:** Factor the expression $$12x^5y^4 - 9xy^5 + 27x^2y^8$$ by factoring out the greatest common factor (GCF).
2. **Find the GCF of the coefficients:** The coefficients are 12, 9, and 27. The GCF of 12, 9, and 27 is 3.
3. **Find the GCF of the variables:**
- For $$x^5, x^1, x^2$$ the smallest power is $$x^1$$.
- For $$y^4, y^5, y^8$$ the smallest power is $$y^4$$.
4. **Therefore, the GCF is:** $$3xy^4$$.
5. **Divide each term by the GCF:**
- $$\frac{12x^5y^4}{3xy^4} = 4x^{5-1}y^{4-4} = 4x^4$$
- $$\frac{-9xy^5}{3xy^4} = -3x^{1-1}y^{5-4} = -3y$$
- $$\frac{27x^2y^8}{3xy^4} = 9x^{2-1}y^{8-4} = 9xy^4$$
6. **Write the factored form:**
$$3xy^4(4x^4 - 3y + 9xy^4)$$
7. **Check if the factorization is complete:** The expression inside the parentheses cannot be factored further.
8. **Verify the answer:** The given answer matches the correct factorization.
**Final answer:** $$3xy^4(4x^4 - 3y + 9xy^4)$$