To determine the greatest common factor (GCF) of the given terms [tex]\(42 a^5 b^3\)[/tex], [tex]\(35 a^3 b^4\)[/tex], and [tex]\(42 a b^4\)[/tex], let's break the problem into manageable steps.
1. Find the Greatest Common Factor of the Coefficients:
- The coefficients of the given terms are 42, 35, and 42.
- The GCF of these coefficients can be determined by finding the highest number that divides all three coefficients without leaving a remainder.
The prime factorization of the coefficients:
- [tex]\(42 = 2 \times 3 \times 7\)[/tex]
- [tex]\(35 = 5 \times 7\)[/tex]
- [tex]\(42 = 2 \times 3 \times 7\)[/tex]
The common factor among these coefficients is [tex]\(7\)[/tex], as 7 is the highest number that divides all three.
2. Determine the Exponents for 'a':
- The exponents of [tex]\(a\)[/tex] in the terms are 5, 3, and 1.
- The minimum exponent common to all terms is the smallest one.
- The smallest exponent of [tex]\(a\)[/tex] among 5, 3, and 1 is 1.
3. Determine the Exponents for 'b':
- The exponents of [tex]\(b\)[/tex] in the terms are 3, 4, and 4.
- The minimum exponent common to all terms is the smallest one.
- The smallest exponent of [tex]\(b\)[/tex] among 3, 4, and 4 is 3.
Combining the above results, the GCF of the three terms is given by:
- The GCF of the coefficients: [tex]\(7\)[/tex]
- The minimum exponent for [tex]\(a\)[/tex]: [tex]\(a^1\)[/tex] (written as [tex]\(a\)[/tex])
- The minimum exponent for [tex]\(b\)[/tex]: [tex]\(b^3\)[/tex]
Thus, the Greatest Common Factor of [tex]\(42 a^5 b^3\)[/tex], [tex]\(35 a^3 b^4\)[/tex], and [tex]\(42 a b^4\)[/tex] is [tex]\(7 a b^3\)[/tex].
Therefore, the answer is:
[tex]\[ \boxed{7 a b^3} \][/tex]
