To find the greatest common factor (GCF) of the expressions [tex]\(6x^5\)[/tex] and [tex]\(12x^4\)[/tex], we need to determine both the GCF of the numerical coefficients and the GCF of the variable parts separately. Here’s a step-by-step breakdown:
1. Identify the coefficients and the variable parts in each term:
- For [tex]\(6x^5\)[/tex]:
- Coefficient: 6
- Variable part: [tex]\(x^5\)[/tex]
- For [tex]\(12x^4\)[/tex]:
- Coefficient: 12
- Variable part: [tex]\(x^4\)[/tex]
2. Find the GCF of the coefficients:
- The coefficients are 6 and 12.
- The prime factorizations are:
- [tex]\(6 = 2 \times 3\)[/tex]
- [tex]\(12 = 2^2 \times 3\)[/tex]
- The common prime factors with the smallest power are:
- [tex]\(2\)[/tex]
- [tex]\(3\)[/tex]
- Therefore, the GCF of 6 and 12 is:
- [tex]\(2 \times 3 = 6\)[/tex]
3. Find the GCF of the variable parts:
- The variable parts are [tex]\(x^5\)[/tex] and [tex]\(x^4\)[/tex].
- The GCF of [tex]\(x^5\)[/tex] and [tex]\(x^4\)[/tex] is determined by the lowest power of [tex]\(x\)[/tex] common to both terms, which is [tex]\(x^4\)[/tex].
4. Combine the GCF of the coefficients and the variables:
- The GCF of the coefficients is 6.
- The GCF of the variable parts is [tex]\(x^4\)[/tex].
- Therefore, the GCF of the entire terms [tex]\(6x^5\)[/tex] and [tex]\(12x^4\)[/tex] is:
- [tex]\(6x^4\)[/tex]
Thus, the greatest common factor of [tex]\(6x^5\)[/tex] and [tex]\(12x^4\)[/tex] is [tex]\(6x^4\)[/tex].
The correct answer is:
[tex]\[ \boxed{6x^4} \][/tex]
