Answer :
To find the greatest common factor (GCF) of the expressions [tex]\(10x^5\)[/tex], [tex]\(35x^4\)[/tex], and [tex]\(2x^2\)[/tex], follow these steps:
1. Factor the Coefficients:
- The coefficients of the expressions are 10, 35, and 2.
- Factor each:
- [tex]\(10 = 2 \times 5\)[/tex]
- [tex]\(35 = 5 \times 7\)[/tex]
- [tex]\(2 = 2\)[/tex]
- The greatest common factor of 10, 35, and 2 is 1, since these numbers have no common prime factor besides 1.
2. Determine the Lowest Power of [tex]\(x\)[/tex]:
- The powers of [tex]\(x\)[/tex] in the expressions are [tex]\(x^5\)[/tex], [tex]\(x^4\)[/tex], and [tex]\(x^2\)[/tex].
- To find the GCF of expressions with powers of a variable, choose the variable raised to the lowest power common among all expressions.
- Among the powers 5, 4, and 2, the smallest is 2.
3. Combine the Results:
- Combine the greatest common factor of the coefficients and the lowest common power of the variable [tex]\(x\)[/tex].
- The GCF of the expressions is [tex]\(x^2\)[/tex], since we have a common factor of 1 for the coefficients and [tex]\(x^2\)[/tex] for the variable.
Therefore, the greatest common factor of the expressions [tex]\(10x^5\)[/tex], [tex]\(35x^4\)[/tex], and [tex]\(2x^2\)[/tex] is [tex]\(x^2\)[/tex].
1. Factor the Coefficients:
- The coefficients of the expressions are 10, 35, and 2.
- Factor each:
- [tex]\(10 = 2 \times 5\)[/tex]
- [tex]\(35 = 5 \times 7\)[/tex]
- [tex]\(2 = 2\)[/tex]
- The greatest common factor of 10, 35, and 2 is 1, since these numbers have no common prime factor besides 1.
2. Determine the Lowest Power of [tex]\(x\)[/tex]:
- The powers of [tex]\(x\)[/tex] in the expressions are [tex]\(x^5\)[/tex], [tex]\(x^4\)[/tex], and [tex]\(x^2\)[/tex].
- To find the GCF of expressions with powers of a variable, choose the variable raised to the lowest power common among all expressions.
- Among the powers 5, 4, and 2, the smallest is 2.
3. Combine the Results:
- Combine the greatest common factor of the coefficients and the lowest common power of the variable [tex]\(x\)[/tex].
- The GCF of the expressions is [tex]\(x^2\)[/tex], since we have a common factor of 1 for the coefficients and [tex]\(x^2\)[/tex] for the variable.
Therefore, the greatest common factor of the expressions [tex]\(10x^5\)[/tex], [tex]\(35x^4\)[/tex], and [tex]\(2x^2\)[/tex] is [tex]\(x^2\)[/tex].
