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Answer
The expression \(10 p^{3} + 2 s^{2} y - 8 r y^{2}\) can be factored as \(2(5 p^{3} + s^{2} y - 4 r y^{2})\).
Solution
To factorize the expression \( 10 p^{3} + 2 s^{2} y - 8 r y^{2} \), follow these steps:
1. **Identify the Greatest Common Factor (GCF):**
- Look for the highest common factor among the coefficients \(10\), \(2\), and \(-8\).
- The GCF of \(10\), \(2\), and \(-8\) is \(2\).
2. **Factor Out the GCF:**
\[
10 p^{3} + 2 s^{2} y - 8 r y^{2} = 2 \left(5 p^{3} + s^{2} y - 4 r y^{2}\right)
\]
3. **Check for Further Factorization:**
- Examine the expression inside the parentheses: \(5 p^{3} + s^{2} y - 4 r y^{2}\).
- There are no common factors among the terms \(5 p^{3}\), \(s^{2} y\), and \(-4 r y^{2}\).
- Additionally, since the variables \(p\), \(s\), \(r\), and \(y\) are distinct, the expression does not factor further using standard factoring techniques.
**Final Factorized Form:**
\[
2 \left(5 p^{3} + s^{2} y - 4 r y^{2}\right)
\]
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Simplify this solution Extra Insights
To factor the expression \( 10 p^{3} + 2 s^{2} y - 8 r y^{2} \), we first look for common factors. Notice that the first two terms share a factor of 2. Factoring out 2, we get: \[ 2(5 p^{3} + s^{2} y - 4 ry^{2}) \] The remaining expression \( 5 p^{3} + s^{2} y - 4 ry^{2} \) does not have common factors. This can be assessed further or potentially combined, but as it stands, the primary factorization step is complete. So, the expression can be written as: \[ 2(5 p^{3} + s^{2} y - 4 r y^{2}) \]
