To find the greatest common factor (GCF) of the monomials [tex]\(12x^5\)[/tex] and [tex]\(8x^3\)[/tex], follow these steps:
1. Identify the coefficients and powers of [tex]\(x\)[/tex] in each monomial:
- For [tex]\(12x^5\)[/tex]: The coefficient is 12 and the power of [tex]\(x\)[/tex] is 5.
- For [tex]\(8x^3\)[/tex]: The coefficient is 8 and the power of [tex]\(x\)[/tex] is 3.
2. Determine the greatest common factor (GCF) of the coefficients:
- The GCF of 12 and 8 can be found by considering their prime factorizations:
- [tex]\(12 = 2^2 \times 3\)[/tex]
- [tex]\(8 = 2^3\)[/tex]
- The common factors are [tex]\(2^2\)[/tex]. Thus, the GCF of 12 and 8 is [tex]\(2^2 = 4\)[/tex].
3. Determine the GCF of the powers of [tex]\(x\)[/tex]:
- The powers of [tex]\(x\)[/tex] are 5 (from [tex]\(12x^5\)[/tex]) and 3 (from [tex]\(8x^3\)[/tex]).
- The GCF of these powers is the smaller of the two, which is [tex]\( \min(5, 3) = 3 \)[/tex].
4. Combine the GCF of the coefficients and the GCF of the powers of [tex]\(x\)[/tex]:
- The GCF of the coefficients is 4.
- The GCF of the powers of [tex]\(x\)[/tex] is [tex]\(x^3\)[/tex].
Therefore, the greatest common factor of [tex]\(12x^5\)[/tex] and [tex]\(8x^3\)[/tex] is:
[tex]\[ 4x^3 \][/tex]
