For the given polynomial `p (x) = x ^(2) - 5x -1,`if `alpha and beta` are its zeroes then find the value of `alpha ^(2) beta + alpha beta ^(2)`

Options

A. `-5`
B. 4
C. 0
D. `-7`

Answer

To find the value of \( \alpha^2 \beta + \alpha \beta^2 \) for the polynomial \( p(x) = x^2 - 5x - 1 \), where \( \alpha \) and \( \beta \) are its zeroes, we can follow these steps:

### Step 1: Identify the coefficients of the polynomial
The polynomial is given as \( p(x) = x^2 - 5x - 1 \). Here, we can identify:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = -5 \) (coefficient of \( x \))
- \( c = -1 \) (constant term)

### Step 2: Use Vieta's formulas to find \( \alpha + \beta \) and \( \alpha \beta \)
From Vieta's formulas, we know:
- \( \alpha + \beta = -\frac{b}{a} = -\frac{-5}{1} = 5 \)
- \( \alpha \beta = \frac{c}{a} = \frac{-1}{1} = -1 \)

### Step 3: Express \( \alpha^2 \beta + \alpha \beta^2 \) in terms of \( \alpha + \beta \) and \( \alpha \beta \)
We can factor \( \alpha^2 \beta + \alpha \beta^2 \) as follows:
\[ \alpha^2 \beta + \alpha \beta^2 = \alpha \beta (\alpha + \beta) \]

### Step 4: Substitute the values of \( \alpha + \beta \) and \( \alpha \beta \)
Now we can substitute the values we found:
\[ \alpha^2 \beta + \alpha \beta^2 = \alpha \beta (\alpha + \beta) = (-1)(5) = -5 \]

### Final Answer
Thus, the value of \( \alpha^2 \beta + \alpha \beta^2 \) is \( -5 \).

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