What is the greatest common factor of the polynomial [tex]21x^4 + 15x^3 + 6x^2[/tex]?

To find the greatest common factor (GCF) of the polynomial [tex]\(21x^4 + 15x^3 + 6x^2\)[/tex], we can follow these steps:

1. Identify the coefficients and variables:
- The polynomial is made up of the terms: [tex]\(21x^4\)[/tex], [tex]\(15x^3\)[/tex], and [tex]\(6x^2\)[/tex].
- The coefficients of these terms are 21, 15, and 6, respectively.
- The variables include powers of [tex]\(x\)[/tex], specifically [tex]\(x^4\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^2\)[/tex].

2. Find the GCF of the coefficients:
- List the factors of each coefficient:
- Factors of 21: 1, 3, 7, 21
- Factors of 15: 1, 3, 5, 15
- Factors of 6: 1, 2, 3, 6
- The greatest common factor among these numbers is 3.

3. Find the GCF of the variables:
- Look at the powers of [tex]\(x\)[/tex]: [tex]\(x^4\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^2\)[/tex].
- The lowest power is [tex]\(x^2\)[/tex].

4. Combine the GCFs:
- The greatest common factor of the numerical coefficients is 3.
- The greatest common factor of the variable parts is [tex]\(x^2\)[/tex].

Therefore, the greatest common factor of the polynomial [tex]\(21x^4 + 15x^3 + 6x^2\)[/tex] is [tex]\(3x^2\)[/tex].