To determine the greatest common factor (GCF) of the given terms [tex]\(5 x^7\)[/tex], [tex]\(30 x^4\)[/tex], and [tex]\(10 x^3\)[/tex], we need to follow these steps:
1. Factor each term into its prime factors and powers of [tex]\(x\)[/tex]:
- [tex]\(5 x^7\)[/tex] has constants and variables:
- [tex]\(5\)[/tex] as the coefficient (which is a prime number)
- [tex]\(x^7\)[/tex]
- [tex]\(30 x^4\)[/tex] can be broken down as:
- [tex]\(30\)[/tex] is composed of [tex]\(2 \times 3 \times 5\)[/tex]
- [tex]\(x^4\)[/tex]
- [tex]\(10 x^3\)[/tex] can be broken down as:
- [tex]\(10\)[/tex] is composed of [tex]\(2 \times 5\)[/tex]
- [tex]\(x^3\)[/tex]
2. Identify the common factors:
We need to find the common factors in both the constants (coefficients) and the variables:
- Constants:
- The constants we have are [tex]\(5\)[/tex], [tex]\(30\)[/tex], and [tex]\(10\)[/tex].
- The factors of these constants:
- The factor of [tex]\(5\)[/tex] is [tex]\(5\)[/tex]
- The factors of [tex]\(30\)[/tex] are [tex]\(2, 3,\)[/tex] and [tex]\(5\)[/tex]
- The factors of [tex]\(10\)[/tex] are [tex]\(2\)[/tex] and [tex]\(5\)[/tex]
- The common factor among [tex]\(5\)[/tex], [tex]\(30\)[/tex], and [tex]\(10\)[/tex] is [tex]\(5\)[/tex].
- Variables:
- The variables are [tex]\(x^7\)[/tex], [tex]\(x^4\)[/tex], and [tex]\(x^3\)[/tex].
- The powers of [tex]\(x\)[/tex] are [tex]\(7\)[/tex], [tex]\(4\)[/tex], and [tex]\(3\)[/tex].
- The lowest power of [tex]\(x\)[/tex] common to all the expressions is [tex]\(x^3\)[/tex].
3. Combine the common factors:
To find the greatest common factor, combine the common factors of the coefficients and the lowest power of the variables:
- The common factor of the coefficients is [tex]\(5\)[/tex].
- The common power of the variable [tex]\(x\)[/tex] is [tex]\(x^3\)[/tex].
Therefore, combining these, we get the greatest common factor (GCF):
[tex]\[ \boxed{5 x^3} \][/tex]
This completes the solution: the greatest common factor of [tex]\(5 x^7\)[/tex], [tex]\(30 x^4\)[/tex], and [tex]\(10 x^3\)[/tex] is [tex]\(5 x^3\)[/tex].
