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Answer :
Sure, let's factor the greatest common factor (GCF) out of the polynomial [tex]\(35x^5 + 15x^3 + 5x^2\)[/tex].
### Step-by-step Solution:
1. Identify the coefficients of the terms: In the polynomial [tex]\(35x^5 + 15x^3 + 5x^2\)[/tex], the coefficients are:
- 35 (from [tex]\(35x^5\)[/tex])
- 15 (from [tex]\(15x^3\)[/tex])
- 5 (from [tex]\(5x^2\)[/tex])
2. Find the GCF of the coefficients:
- The GCF of the coefficients 35, 15, and 5 can be determined by finding the greatest number that divides all three coefficients evenly.
- The greatest common divisor (GCD) of 35, 15, and 5 is 5.
3. Factor out the GCF from each term:
- Divide each coefficient by the GCF (5 in this case):
- [tex]\( \frac{35}{5} = 7 \)[/tex]
- [tex]\( \frac{15}{5} = 3 \)[/tex]
- [tex]\( \frac{5}{5} = 1 \)[/tex]
4. Rewrite each term with the GCF factored out:
- For [tex]\(35x^5\)[/tex], when factored by 5, it becomes [tex]\(5 \cdot 7x^5\)[/tex].
- For [tex]\(15x^3\)[/tex], when factored by 5, it becomes [tex]\(5 \cdot 3x^3\)[/tex].
- For [tex]\(5x^2\)[/tex], when factored by 5, it becomes [tex]\(5 \cdot 1x^2\)[/tex].
5. Express the polynomial with the GCF factored out:
- Combining these, we have the polynomial rewritten as [tex]\( 5(7x^5 + 3x^3 + 1x^2) \)[/tex].
Therefore, the polynomial [tex]\(35x^5 + 15x^3 + 5x^2\)[/tex] factored out by the GCF is:
[tex]\[
\boxed{5(7x^5 + 3x^3 + x^2)}
\][/tex]
This is the fully factored form of the given polynomial.
### Step-by-step Solution:
1. Identify the coefficients of the terms: In the polynomial [tex]\(35x^5 + 15x^3 + 5x^2\)[/tex], the coefficients are:
- 35 (from [tex]\(35x^5\)[/tex])
- 15 (from [tex]\(15x^3\)[/tex])
- 5 (from [tex]\(5x^2\)[/tex])
2. Find the GCF of the coefficients:
- The GCF of the coefficients 35, 15, and 5 can be determined by finding the greatest number that divides all three coefficients evenly.
- The greatest common divisor (GCD) of 35, 15, and 5 is 5.
3. Factor out the GCF from each term:
- Divide each coefficient by the GCF (5 in this case):
- [tex]\( \frac{35}{5} = 7 \)[/tex]
- [tex]\( \frac{15}{5} = 3 \)[/tex]
- [tex]\( \frac{5}{5} = 1 \)[/tex]
4. Rewrite each term with the GCF factored out:
- For [tex]\(35x^5\)[/tex], when factored by 5, it becomes [tex]\(5 \cdot 7x^5\)[/tex].
- For [tex]\(15x^3\)[/tex], when factored by 5, it becomes [tex]\(5 \cdot 3x^3\)[/tex].
- For [tex]\(5x^2\)[/tex], when factored by 5, it becomes [tex]\(5 \cdot 1x^2\)[/tex].
5. Express the polynomial with the GCF factored out:
- Combining these, we have the polynomial rewritten as [tex]\( 5(7x^5 + 3x^3 + 1x^2) \)[/tex].
Therefore, the polynomial [tex]\(35x^5 + 15x^3 + 5x^2\)[/tex] factored out by the GCF is:
[tex]\[
\boxed{5(7x^5 + 3x^3 + x^2)}
\][/tex]
This is the fully factored form of the given polynomial.
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