Question
Factor out the greatest common factor in the expression. \( 5 x^{4}-15 x^{3}+25 x^{2} \) Select the correct choice below and fill in any answer boxes within your choice. A. \( 5 x^{4}-15 x^{3}+25 x^{2}=\square \) B. The polynomial is prime.
Ask by Hilton West. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( 5x^{4} - 15x^{3} + 25x^{2} = 5x^{2}(x^{2} - 3x + 5) \)
Solution
To factor out the greatest common factor (GCF) from the expression \( 5x^{4} - 15x^{3} + 25x^{2} \), follow these steps:
1. **Identify the GCF of the coefficients:**
- The coefficients are 5, -15, and 25.
- The GCF of 5, 15, and 25 is **5**.
2. **Identify the GCF of the variable parts:**
- The variable parts are \( x^{4} \), \( x^{3} \), and \( x^{2} \).
- The lowest exponent is \( x^{2} \).
3. **Combine the GCFs:**
- The overall GCF is \( 5x^{2} \).
4. **Factor out the GCF:**
\[
5x^{4} - 15x^{3} + 25x^{2} = 5x^{2}(x^{2} - 3x + 5)
\]
So, the correct choice is:
**A.**
\[
5x^{4} - 15x^{3} + 25x^{2} = 5x^{2}(x^{2} - 3x + 5)
\]
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Simplify this solution Extra Insights
To factor out the greatest common factor from the expression \( 5 x^{4}-15 x^{3}+25 x^{2} \), we first identify the GCF of the coefficients (5, -15, and 25) and the smallest power of \( x \) present in all terms, which is \( x^{2} \). The GCF is \( 5x^{2} \). Now, we can factor \( 5x^{2} \) out of the expression: \[ 5 x^{4}-15 x^{3}+25 x^{2} = 5 x^{2}(x^{2}-3x+5) \] So the correct choice is: A. \( 5 x^{4}-15 x^{3}+25 x^{2} = 5 x^{2}(x^{2}-3x+5) \)
