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Answer :
To find the greatest common factor (GCF) of the polynomial [tex]\(25x^4 + 35x^3 + 5x^2\)[/tex], we should follow these steps:
1. Identify the coefficients of the polynomial:
The coefficients are the numerical parts in front of each term involving [tex]\(x\)[/tex]. For this polynomial, the coefficients are 25, 35, and 5.
2. Find the GCF of the coefficients:
- List the factors for each coefficient:
- Factors of 25: 1, 5, 25
- Factors of 35: 1, 5, 7, 35
- Factors of 5: 1, 5
- The greatest common factor among 25, 35, and 5 is 5.
3. Identify the lowest power of [tex]\(x\)[/tex] in each term:
Each term has a power of [tex]\(x\)[/tex]. The powers are:
- [tex]\(x^4\)[/tex] in the term [tex]\(25x^4\)[/tex]
- [tex]\(x^3\)[/tex] in the term [tex]\(35x^3\)[/tex]
- [tex]\(x^2\)[/tex] in the term [tex]\(5x^2\)[/tex]
The lowest power of [tex]\(x\)[/tex] that is common to all terms is [tex]\(x^2\)[/tex].
4. Combine the results:
The greatest common factor of the polynomial is the product of the GCF of the coefficients and the lowest power of [tex]\(x\)[/tex] found. Therefore, the GCF is [tex]\(5x^2\)[/tex].
The greatest common factor of the polynomial [tex]\(25x^4 + 35x^3 + 5x^2\)[/tex] is [tex]\(5x^2\)[/tex].
1. Identify the coefficients of the polynomial:
The coefficients are the numerical parts in front of each term involving [tex]\(x\)[/tex]. For this polynomial, the coefficients are 25, 35, and 5.
2. Find the GCF of the coefficients:
- List the factors for each coefficient:
- Factors of 25: 1, 5, 25
- Factors of 35: 1, 5, 7, 35
- Factors of 5: 1, 5
- The greatest common factor among 25, 35, and 5 is 5.
3. Identify the lowest power of [tex]\(x\)[/tex] in each term:
Each term has a power of [tex]\(x\)[/tex]. The powers are:
- [tex]\(x^4\)[/tex] in the term [tex]\(25x^4\)[/tex]
- [tex]\(x^3\)[/tex] in the term [tex]\(35x^3\)[/tex]
- [tex]\(x^2\)[/tex] in the term [tex]\(5x^2\)[/tex]
The lowest power of [tex]\(x\)[/tex] that is common to all terms is [tex]\(x^2\)[/tex].
4. Combine the results:
The greatest common factor of the polynomial is the product of the GCF of the coefficients and the lowest power of [tex]\(x\)[/tex] found. Therefore, the GCF is [tex]\(5x^2\)[/tex].
The greatest common factor of the polynomial [tex]\(25x^4 + 35x^3 + 5x^2\)[/tex] is [tex]\(5x^2\)[/tex].
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