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Answer :
To find the greatest common factor (GCF) of the terms in the given polynomial list, we first need to identify the common factors among the coefficients and the variables.
Let's break down each polynomial:
1. 35x^5 + 25x^4 + 10x^3:
- Coefficients: 35, 25, 10
- The GCF of the coefficients is 5, since 5 is the largest number that divides all of them.
- The smallest power of x is x^3.
2. 5x^5:
- Coefficient: 5
- Variable: x^5
3. 5x:
- Coefficient: 5
- Variable: x
4. 5x^3:
- Coefficient: 5
- Variable: x^3
5. x^3:
- Coefficient: Implicitly 1
- Variable: x^3
Now let's determine the overall greatest common factor:
- For the coefficients of these terms (35, 25, 10, 5, 5, 5, 1), the GCF is 5.
- For the variable part, the smallest power of x that appears in all terms is x^3.
Putting it all together, the greatest common factor is the product of the GCF of the coefficients and the smallest power of x:
[tex]\[
\text{GCF} = 5x^3
\][/tex]
Thus, the greatest common factor of the terms from the listed polynomials is [tex]\(5x^3\)[/tex].
Let's break down each polynomial:
1. 35x^5 + 25x^4 + 10x^3:
- Coefficients: 35, 25, 10
- The GCF of the coefficients is 5, since 5 is the largest number that divides all of them.
- The smallest power of x is x^3.
2. 5x^5:
- Coefficient: 5
- Variable: x^5
3. 5x:
- Coefficient: 5
- Variable: x
4. 5x^3:
- Coefficient: 5
- Variable: x^3
5. x^3:
- Coefficient: Implicitly 1
- Variable: x^3
Now let's determine the overall greatest common factor:
- For the coefficients of these terms (35, 25, 10, 5, 5, 5, 1), the GCF is 5.
- For the variable part, the smallest power of x that appears in all terms is x^3.
Putting it all together, the greatest common factor is the product of the GCF of the coefficients and the smallest power of x:
[tex]\[
\text{GCF} = 5x^3
\][/tex]
Thus, the greatest common factor of the terms from the listed polynomials is [tex]\(5x^3\)[/tex].
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