To determine the greatest common factor (GCF) of the given terms [tex]\(14 x^2 y^3\)[/tex] and [tex]\(21 x y^2\)[/tex], follow these steps:
1. Identify the coefficients and variables:
- The first term is [tex]\(14 x^2 y^3\)[/tex]:
- Coefficient: 14
- Variables: [tex]\(x^2\)[/tex] and [tex]\(y^3\)[/tex]
- The second term is [tex]\(21 x y^2\)[/tex]:
- Coefficient: 21
- Variables: [tex]\(x^1\)[/tex] and [tex]\(y^2\)[/tex]
2. Find the GCF of the coefficients:
- The coefficients are 14 and 21.
- The greatest common factor of 14 and 21 is determined by finding the largest integer that divides both 14 and 21 without leaving a remainder.
- The GCF of 14 and 21 is 7.
3. Determine the GCF for each variable by comparing their exponents:
- For [tex]\(x\)[/tex]:
- The exponents are 2 (from [tex]\(14 x^2 y^3\)[/tex]) and 1 (from [tex]\(21 x y^2\)[/tex]).
- The GCF of [tex]\(x^2\)[/tex] and [tex]\(x^1\)[/tex] is [tex]\(x^1\)[/tex] since we take the smaller exponent.
- For [tex]\(y\)[/tex]:
- The exponents are 3 (from [tex]\(14 x^2 y^3\)[/tex]) and 2 (from [tex]\(21 x y^2\)[/tex]).
- The GCF of [tex]\(y^3\)[/tex] and [tex]\(y^2\)[/tex] is [tex]\(y^2\)[/tex] since we take the smaller exponent.
4. Combine the GCFs:
- Coefficient: 7
- Variables: [tex]\(x^1\)[/tex] and [tex]\(y^2\)[/tex]
Therefore, the greatest common factor (GCF) of [tex]\(14 x^2 y^3\)[/tex] and [tex]\(21 x y^2\)[/tex] is:
[tex]\[ 7 x^1 y^2 \][/tex]
Simplifying further, we get:
[tex]\[ 7 x y^2 \][/tex]
So, the GCF of [tex]\(14 x^2 y^3\)[/tex] and [tex]\(21 x y^2\)[/tex] is [tex]\(7 x y^2\)[/tex].
