If `alpha and beta` be the zeros of the polynomial `x^(2)+x+1`, 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 \( x^2 + x + 1 \), we can follow these steps:

### Step 1: Identify the coefficients of the polynomial
The polynomial is given as \( x^2 + x + 1 \). Here, we can identify:
- \( a = 1 \)
- \( b = 1 \)
- \( c = 1 \)

### Step 2: Use Vieta's formulas to find the sum and product of the roots
According to Vieta's formulas:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} \)

Substituting the values of \( a \), \( b \), and \( c \):
- \( \alpha + \beta = -\frac{1}{1} = -1 \)
- \( \alpha \beta = \frac{1}{1} = 1 \)

### Step 3: Calculate \( \frac{1}{\alpha} + \frac{1}{\beta} \)
We can rewrite \( \frac{1}{\alpha} + \frac{1}{\beta} \) as:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta + \alpha}{\alpha \beta} \]

Substituting the values we found:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta} = \frac{-1}{1} = -1 \]

### Final Answer
Thus, the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \) is \( -1 \).

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