To factor the Greatest Common Factor (GCF) out of the polynomial $21x^6y^3 + 6x^5y^4 + 3x^4y^2$, follow these steps:

1. Identify the GCF of the coefficients:

   • The coefficients are $21$, $6$, and $3$.
   • The GCF of these numbers is $3$.

2. Identify the GCF of the variables $x$:

   • The terms contain $x^6$, $x^5$, and $x^4$.
   • The GCF of these terms is $x^4$ (the lowest power of $x$ present in all terms).

3. Identify the GCF of the variables $y$:

   • The terms contain $y^3$, $y^4$, and $y^2$.
   • The GCF of these terms is $y^2$ (the lowest power of $y$ present in all terms).

4. Combine the GCF of the coefficients and variables:

   • The overall GCF of the polynomial is $3x^4y^2$.

5. Factor the GCF out of each term:

   • Factor $3x^4y^2$ out of $21x^6y^3$: $21x^6y^3 \div 3x^4y^2 = 7x^2y$.
   • Factor $3x^4y^2$ out of $6x^5y^4$: $6x^5y^4 \div 3x^4y^2 = 2xy^2$.
   • Factor $3x^4y^2$ out of $3x^4y^2$: $3x^4y^2 \div 3x^4y^2 = 1$.

6. Write the factored expression:

   • The factored form of the polynomial is: $21x^6y^3 + 6x^5y^4 + 3x^4y^2 = 3x^4y^2(7x^2y + 2xy^2 + 1)$

So, the final factored expression is:

$3x^4y^2(7x^2y + 2xy^2 + 1)$

Would you like any more details or have any questions about this process?

---

Here are 5 related questions:

1. How do you determine the GCF when dealing with multiple variables in a polynomial?
2. What is the next step after factoring the GCF if the polynomial needs to be factored completely?
3. How do you factor polynomials with different exponents for each variable?
4. What happens if the GCF is just a variable and not a constant?
5. Can you factor out the GCF from a polynomial if one of the terms lacks a variable?

Tip: When factoring out the GCF, make sure to include all parts of the polynomial in the GCF, considering both coefficients and variable terms.