To find the greatest common factor (GCF) of the given terms [tex]\(42 a^5 b^3\)[/tex], [tex]\(36 a^3 b^4\)[/tex], and [tex]\(42 a b^4\)[/tex], we need to determine the common factor in terms of both coefficients and variables.
### 1. Coefficients:
The coefficients of the terms are [tex]\(42\)[/tex], [tex]\(36\)[/tex], and [tex]\(42\)[/tex].
- We need to find the GCF of these numbers:
- The prime factorization of [tex]\(42\)[/tex] is [tex]\(2 \times 3 \times 7\)[/tex].
- The prime factorization of [tex]\(36\)[/tex] is [tex]\(2^2 \times 3^2\)[/tex].
- Again, the prime factorization of [tex]\(42\)[/tex] is [tex]\(2 \times 3 \times 7\)[/tex].
- The common prime factors are [tex]\(2\)[/tex] and [tex]\(3\)[/tex], each raised to the lowest power found in each factorization:
- For [tex]\(2\)[/tex], the lowest power is [tex]\(2^1\)[/tex].
- For [tex]\(3\)[/tex], the lowest power is [tex]\(3^1\)[/tex].
- Thus, the numerical GCF:
[tex]\[
2 \times 3 = 6
\][/tex]
### 2. Variables:
Now, we analyze the variables [tex]\(a\)[/tex] and [tex]\(b\)[/tex] with their respective exponents.
- For [tex]\(a\)[/tex]:
- In [tex]\(42 a^5 b^3\)[/tex], the exponent of [tex]\(a\)[/tex] is [tex]\(5\)[/tex].
- In [tex]\(36 a^3 b^4\)[/tex], the exponent of [tex]\(a\)[/tex] is [tex]\(3\)[/tex].
- In [tex]\(42 a b^4\)[/tex], the exponent of [tex]\(a\)[/tex] is [tex]\(1\)[/tex].
The minimum exponent of [tex]\(a\)[/tex] is [tex]\(1\)[/tex].
- For [tex]\(b\)[/tex]:
- In [tex]\(42 a^5 b^3\)[/tex], the exponent of [tex]\(b\)[/tex] is [tex]\(3\)[/tex].
- In [tex]\(36 a^3 b^4\)[/tex], the exponent of [tex]\(b\)[/tex] is [tex]\(4\)[/tex].
- In [tex]\(42 a b^4\)[/tex], the exponent of [tex]\(b\)[/tex] is [tex]\(4\)[/tex].
The minimum exponent of [tex]\(b\)[/tex] is [tex]\(3\)[/tex].
### 3. Combine the factors:
Putting it all together, the GCF is the product of the numerical GCF and the variable factors with their minimum exponents:
[tex]\[
\text{GCF} = 6 \times a^1 \times b^3 = 6a b^3
\][/tex]
Thus, the greatest common factor of [tex]\(42 a^5 b^3\)[/tex], [tex]\(36 a^3 b^4\)[/tex], and [tex]\(42 a b^4\)[/tex] is:
[tex]\[
\boxed{6ab^3}
\][/tex]
Therefore, none of the given options are correct.
