A cubic polynomial with sum of its zeroes, sum of the product of its zeroes taken two at a time and the product of its zeroes as -3, 8, 4 respectively, is_____.

Options

A. `x^(3) - 3x^(2) - 8x-4`
B. `x^(3) + 3x^(2) - 8x - 4`
C. `x^(3) + 3x^(2) + 8x-4`
D. `x^(3) -3x^(2) -8x+4`

Answer

To find the cubic polynomial given the sum of its zeroes, the sum of the product of its zeroes taken two at a time, and the product of its zeroes, we can follow these steps:

### Step 1: Identify the given values
We are given:
- Sum of the zeroes (α + β + γ) = -3
- Sum of the product of the zeroes taken two at a time (αβ + βγ + γα) = 8
- Product of the zeroes (αβγ) = 4

### Step 2: Write the general form of a cubic polynomial
The general form of a cubic polynomial can be expressed as:
\[ P(x) = x^3 - (α + β + γ)x^2 + (αβ + βγ + γα)x - (αβγ) \]

### Step 3: Substitute the values into the polynomial
Now, substitute the values we identified in Step 1 into the polynomial:
- For \(α + β + γ\), we substitute -3.
- For \(αβ + βγ + γα\), we substitute 8.
- For \(αβγ\), we substitute 4.

Thus, the polynomial becomes:
\[ P(x) = x^3 - (-3)x^2 + 8x - 4 \]

### Step 4: Simplify the polynomial
Now, simplify the expression:
\[ P(x) = x^3 + 3x^2 + 8x - 4 \]

### Step 5: Write the final answer
The required cubic polynomial is:
\[ P(x) = x^3 + 3x^2 + 8x - 4 \]