Find the value of `1/alpha + 1/beta` if `alpha` and `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: Identify the polynomial
The given polynomial is:
\[ x^2 + x + 1 \]

### Step 2: Use Vieta's Formulas
According to Vieta's formulas for a quadratic polynomial \( ax^2 + bx + c \):
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} \)

Here, \( a = 1 \), \( b = 1 \), and \( c = 1 \).

### Step 3: Calculate the sum of the roots
Using Vieta's formulas:
\[ \alpha + \beta = -\frac{1}{1} = -1 \]

### Step 4: Calculate the product of the roots
Using Vieta's formulas:
\[ \alpha \beta = \frac{1}{1} = 1 \]

### Step 5: Find \( \frac{1}{\alpha} + \frac{1}{\beta} \)
We can express \( \frac{1}{\alpha} + \frac{1}{\beta} \) in terms of \( \alpha + \beta \) and \( \alpha \beta \):
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta + \alpha}{\alpha \beta} \]

### Step 6: Substitute the values
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:
\[ \boxed{-1} \]