If `x+y=25 and x^(2)y^(3)+y^(2)x^(3)=25`, what is the value of xy?

Options

A. 0
B. `pm1`
C. 5
D. 4

Answer

To solve the problem step by step, we start with the given equations:

1. **Given Equations**:
- \( x + y = 25 \) (Equation 1)
- \( x^2 y^3 + y^2 x^3 = 25 \) (Equation 2)

2. **Rearranging Equation 2**:
We can factor out \( xy \) from Equation 2:
\[ x^2 y^3 + y^2 x^3 = xy (xy (x + y)) = xy (xy \cdot 25) \]
This simplifies to:
\[ xy (xy \cdot 25) = 25 \]

3. **Setting Up the Equation**:
Let \( xy = k \). Then we can rewrite the equation as:
\[ k(25k) = 25 \]
Simplifying gives:
\[ 25k^2 = 25 \]

4. **Solving for k**:
Dividing both sides by 25:
\[ k^2 = 1 \]
Taking the square root of both sides, we find:
\[ k = \pm 1 \]

5. **Conclusion**:
Therefore, the possible values of \( xy \) are:
\[ xy = 1 \quad \text{or} \quad xy = -1 \]

Thus, the final answer is:
\[ \text{The value of } xy \text{ is } 1 \text{ or } -1. \]