Find the cubic polynomial with the sum, sum of the products of its zeros taken two at a time, and the products of its zeros as – 3,-8 and 2 respectively.

Answer

To find the cubic polynomial given the sum of its zeros, the sum of the products of its zeros taken two at a time, and the product of its zeros, we can use the standard form of a cubic polynomial:

\[ P(x) = x^3 - (sum \ of \ zeros)x^2 + (sum \ of \ products \ of \ zeros \ taken \ two \ at \ a \ time)x - (product \ of \ zeros) \]

### Step-by-step solution:

1. **Identify the given values:**
- Sum of the zeros (α + β + γ) = -3
- Sum of the products of the zeros taken two at a time (αβ + βγ + γα) = -8
- Product of the zeros (αβγ) = 2

2. **Substitute the values into the polynomial formula:**
\[ P(x) = x^3 - (sum \ of \ zeros)x^2 + (sum \ of \ products \ of \ zeros \ taken \ two \ at \ a \ time)x - (product \ of \ zeros) \]
Substituting the values we have:
\[ P(x) = x^3 - (-3)x^2 + (-8)x - 2 \]

3. **Simplify the polynomial:**
- The first term remains \(x^3\).
- The second term becomes \(+3x^2\).
- The third term becomes \(-8x\).
- The last term is \(-2\).
Thus, we have:
\[ P(x) = x^3 + 3x^2 - 8x - 2 \]

4. **Final polynomial:**
The required cubic polynomial is:
\[ P(x) = x^3 + 3x^2 - 8x - 2 \]