Answer :
To find the greatest common factor (GCF) of [tex]\(19x^7\)[/tex] and [tex]\(3x^5\)[/tex], we can break it down into two main parts: the coefficients and the variable parts.
1. Coefficients:
- The coefficients are 19 and 3. The greatest common factor of 19 and 3 is 1 since 19 and 3 are both prime numbers and do not have any common factors other than 1.
2. Variable Part:
- Look at the powers of [tex]\(x\)[/tex] in each term. We have [tex]\(x^7\)[/tex] in the first term and [tex]\(x^5\)[/tex] in the second term.
- The GCF of the variable part is the lowest power of [tex]\(x\)[/tex] that appears in both expressions. So, between [tex]\(x^7\)[/tex] and [tex]\(x^5\)[/tex], the smallest power is [tex]\(x^5\)[/tex].
Therefore, the greatest common factor of the terms [tex]\(19x^7\)[/tex] and [tex]\(3x^5\)[/tex] is simply the product of the GCF of the coefficients and the variable parts, which is [tex]\(1 \times x^5 = x^5\)[/tex].
From the given options, the correct choice is:
A. [tex]\(x^5\)[/tex]
1. Coefficients:
- The coefficients are 19 and 3. The greatest common factor of 19 and 3 is 1 since 19 and 3 are both prime numbers and do not have any common factors other than 1.
2. Variable Part:
- Look at the powers of [tex]\(x\)[/tex] in each term. We have [tex]\(x^7\)[/tex] in the first term and [tex]\(x^5\)[/tex] in the second term.
- The GCF of the variable part is the lowest power of [tex]\(x\)[/tex] that appears in both expressions. So, between [tex]\(x^7\)[/tex] and [tex]\(x^5\)[/tex], the smallest power is [tex]\(x^5\)[/tex].
Therefore, the greatest common factor of the terms [tex]\(19x^7\)[/tex] and [tex]\(3x^5\)[/tex] is simply the product of the GCF of the coefficients and the variable parts, which is [tex]\(1 \times x^5 = x^5\)[/tex].
From the given options, the correct choice is:
A. [tex]\(x^5\)[/tex]
