If `alpha` and `beta` are the zeros of the polynomial `p(x) = x^(2) - px + q`,
then find the value of `(1)/(alpha)+(1)/(beta)`

Answer

To find the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \) where \( \alpha \) and \( \beta \) are the zeros of the polynomial \( p(x) = x^2 - px + q \), we can follow these steps:

### Step 1: Use the relationship of zeros
The expression \( \frac{1}{\alpha} + \frac{1}{\beta} \) can be rewritten using the formula for the sum of the reciprocals of the roots:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta + \alpha}{\alpha \beta} \]

### Step 2: Identify the sum and product of the roots
From Vieta's formulas, for the polynomial \( p(x) = x^2 - px + q \):
- The sum of the roots \( \alpha + \beta = p \)
- The product of the roots \( \alpha \beta = q \)

### Step 3: Substitute the values into the expression
Now, substituting the values from Vieta's formulas into the expression we derived:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta} = \frac{p}{q} \]

### Final Answer
Thus, the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \) is:
\[ \frac{p}{q} \]

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