What is the greatest common factor of [tex]$3x^4$[/tex], [tex]$15x^3$[/tex], and [tex]$21x^2$[/tex]?

To find the greatest common factor (GCF) of the expressions [tex]$3x^4$[/tex], [tex]$15x^3$[/tex], and [tex]$21x^2$[/tex], follow these steps:

1. Identify the coefficients:
- The coefficients of the terms are 3, 15, and 21.

2. Find the GCF of the coefficients:
- First, list the factors of each coefficient:
- Factors of 3: 1, 3
- Factors of 15: 1, 3, 5, 15
- Factors of 21: 1, 3, 7, 21
- The common factors are 1 and 3, so the greatest common factor of the coefficients is 3.

3. Identify the variable parts:
- For the variable [tex]$x$[/tex], the terms are [tex]$x^4$[/tex], [tex]$x^3$[/tex], and [tex]$x^2$[/tex].

4. Find the smallest power of [tex]$x$[/tex]:
- The powers of [tex]$x$[/tex] in each expression are 4, 3, and 2.
- The smallest power is 2.

5. Combine the GCF of the coefficients with the smallest power of [tex]$x$[/tex]:
- The greatest common factor is the product of the GCF of the coefficients (3) and [tex]$x^2$[/tex].

Therefore, the greatest common factor of the expressions [tex]$3x^4$[/tex], [tex]$15x^3$[/tex], and [tex]$21x^2$[/tex] is [tex]$3x^2$[/tex].