Find the value of `1/alpha`+`1/beta`are the zeroes of the polynomial `x^2` + x + 1.

Options

A. 1
B. 0
C. -1
D. 2

Answer

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

### Step 1: Use the formula for the sum of reciprocals of the roots
The sum of the reciprocals of the roots \( \alpha \) and \( \beta \) can be expressed as:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta + \alpha}{\alpha \beta} \]

### Step 2: Identify coefficients from the polynomial
For the polynomial \( x^2 + x + 1 \), we identify the coefficients:
- \( A = 1 \)
- \( B = 1 \)
- \( C = 1 \)

### Step 3: Calculate \( \alpha + \beta \) and \( \alpha \beta \)
Using Vieta's formulas:
- The sum of the roots \( \alpha + \beta = -\frac{B}{A} = -\frac{1}{1} = -1 \)
- The product of the roots \( \alpha \beta = \frac{C}{A} = \frac{1}{1} = 1 \)

### Step 4: Substitute values into the sum of reciprocals formula
Now, substituting the values we found into the formula:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta} = \frac{-1}{1} = -1 \]

### Step 5: Conclusion
Thus, the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \) is:
\[ \boxed{-1} \]

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