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Answer :
Certainly! Let's factor the greatest common factor (GCF) out of the polynomial [tex]\(25x^5 + 10x^3 + 5x^2\)[/tex].
### Step-by-Step Solution
1. Identify the GCF of the Coefficients:
- The coefficients of the terms are 25, 10, and 5.
- The greatest common factor (GCF) of these numbers is 5.
2. Factor the GCF from Each Term:
- Divide each coefficient by the GCF (5) and factor it out:
- For [tex]\(25x^5\)[/tex], divide 25 by 5 to get 5. So, [tex]\(25x^5 = 5 \times 5x^5\)[/tex].
- For [tex]\(10x^3\)[/tex], divide 10 by 5 to get 2. So, [tex]\(10x^3 = 5 \times 2x^3\)[/tex].
- For [tex]\(5x^2\)[/tex], divide 5 by 5 to get 1. So, [tex]\(5x^2 = 5 \times 1x^2\)[/tex].
3. Write the Factored Expression:
- The polynomial rewritten with the GCF factored out is [tex]\(5(5x^5 + 2x^3 + 1x^2)\)[/tex].
4. Simplify the Factored Polynomial:
- The expression [tex]\(5x^5 + 2x^3 + 1x^2\)[/tex] simplifies to [tex]\(x^2(5x^3 + 2x + 1)\)[/tex].
- Therefore, the completely factored polynomial is [tex]\(5x^2(5x^3 + 2x + 1)\)[/tex].
In conclusion, the given polynomial [tex]\(25x^5 + 10x^3 + 5x^2\)[/tex] can be factored as [tex]\(5x^2(5x^3 + 2x + 1)\)[/tex].
### Step-by-Step Solution
1. Identify the GCF of the Coefficients:
- The coefficients of the terms are 25, 10, and 5.
- The greatest common factor (GCF) of these numbers is 5.
2. Factor the GCF from Each Term:
- Divide each coefficient by the GCF (5) and factor it out:
- For [tex]\(25x^5\)[/tex], divide 25 by 5 to get 5. So, [tex]\(25x^5 = 5 \times 5x^5\)[/tex].
- For [tex]\(10x^3\)[/tex], divide 10 by 5 to get 2. So, [tex]\(10x^3 = 5 \times 2x^3\)[/tex].
- For [tex]\(5x^2\)[/tex], divide 5 by 5 to get 1. So, [tex]\(5x^2 = 5 \times 1x^2\)[/tex].
3. Write the Factored Expression:
- The polynomial rewritten with the GCF factored out is [tex]\(5(5x^5 + 2x^3 + 1x^2)\)[/tex].
4. Simplify the Factored Polynomial:
- The expression [tex]\(5x^5 + 2x^3 + 1x^2\)[/tex] simplifies to [tex]\(x^2(5x^3 + 2x + 1)\)[/tex].
- Therefore, the completely factored polynomial is [tex]\(5x^2(5x^3 + 2x + 1)\)[/tex].
In conclusion, the given polynomial [tex]\(25x^5 + 10x^3 + 5x^2\)[/tex] can be factored as [tex]\(5x^2(5x^3 + 2x + 1)\)[/tex].
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