If `alpha,beta` be the roots of `x^(2)-px+q=0`, find the value of `alpha^(5)beta^(7)+alpha^(7)beta^(5)` in terms of p and q.

Answer

To find the value of \( \alpha^5 \beta^7 + \alpha^7 \beta^5 \) in terms of \( p \) and \( q \), where \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( x^2 - px + q = 0 \), we can follow these steps:

### Step 1: Identify the roots and their relationships
From Vieta's formulas, we know:
- The sum of the roots \( \alpha + \beta = p \)
- The product of the roots \( \alpha \beta = q \)

### Step 2: Rewrite the expression
We need to express \( \alpha^5 \beta^7 + \alpha^7 \beta^5 \) in a more manageable form. We can factor it:
\[ \alpha^5 \beta^7 + \alpha^7 \beta^5 = \alpha^5 \beta^5 (\beta^2 + \alpha^2) \]

### Step 3: Simplify \( \beta^2 + \alpha^2 \)
Using the identity \( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \), we can substitute:
\[ \alpha^2 + \beta^2 = p^2 - 2q \]

### Step 4: Calculate \( \alpha^5 \beta^5 \)
We can express \( \alpha^5 \beta^5 \) as:
\[ \alpha^5 \beta^5 = (\alpha \beta)^5 = q^5 \]

### Step 5: Combine the results
Now substituting back into our expression:
\[ \alpha^5 \beta^7 + \alpha^7 \beta^5 = q^5 (p^2 - 2q) \]

### Final Result
Thus, the value of \( \alpha^5 \beta^7 + \alpha^7 \beta^5 \) in terms of \( p \) and \( q \) is:
\[ \alpha^5 \beta^7 + \alpha^7 \beta^5 = q^5 (p^2 - 2q) \]