Answer :
To find the greatest common factor (GCF) of the polynomial [tex]\(25x^5 + 35x^4 + 10x^3\)[/tex], follow these steps:
1. Identify the terms of the polynomial: The polynomial is made up of three terms: [tex]\(25x^5\)[/tex], [tex]\(35x^4\)[/tex], and [tex]\(10x^3\)[/tex].
2. Find the GCF of the coefficients:
- The coefficients here are 25, 35, and 10.
- To find the GCF of these coefficients, list the factors of each:
- Factors of 25: 1, 5, 25
- Factors of 35: 1, 5, 7, 35
- Factors of 10: 1, 2, 5, 10
- The greatest common factor among these is 5.
3. Find the GCF of the variable parts:
- The exponents of [tex]\(x\)[/tex] in the terms are [tex]\(5\)[/tex], [tex]\(4\)[/tex], and [tex]\(3\)[/tex].
- The lowest power of [tex]\(x\)[/tex] that is common in all terms is [tex]\(x^3\)[/tex].
4. Combine the GCF of the coefficients and the variable parts:
- The GCF of the coefficients is 5.
- The GCF of the variable parts is [tex]\(x^3\)[/tex].
- Therefore, the total GCF of the polynomial is [tex]\(5x^3\)[/tex].
So, the greatest common factor of the polynomial [tex]\(25x^5 + 35x^4 + 10x^3\)[/tex] is [tex]\(5x^3\)[/tex].
1. Identify the terms of the polynomial: The polynomial is made up of three terms: [tex]\(25x^5\)[/tex], [tex]\(35x^4\)[/tex], and [tex]\(10x^3\)[/tex].
2. Find the GCF of the coefficients:
- The coefficients here are 25, 35, and 10.
- To find the GCF of these coefficients, list the factors of each:
- Factors of 25: 1, 5, 25
- Factors of 35: 1, 5, 7, 35
- Factors of 10: 1, 2, 5, 10
- The greatest common factor among these is 5.
3. Find the GCF of the variable parts:
- The exponents of [tex]\(x\)[/tex] in the terms are [tex]\(5\)[/tex], [tex]\(4\)[/tex], and [tex]\(3\)[/tex].
- The lowest power of [tex]\(x\)[/tex] that is common in all terms is [tex]\(x^3\)[/tex].
4. Combine the GCF of the coefficients and the variable parts:
- The GCF of the coefficients is 5.
- The GCF of the variable parts is [tex]\(x^3\)[/tex].
- Therefore, the total GCF of the polynomial is [tex]\(5x^3\)[/tex].
So, the greatest common factor of the polynomial [tex]\(25x^5 + 35x^4 + 10x^3\)[/tex] is [tex]\(5x^3\)[/tex].
