Find a cubic polynomial with the sum ,sum of the product of its zeroes taken two at a time ,and the product of its zeroes as 2,-7,-14 respectively.

Answer

To find a cubic polynomial given the sum of its zeroes, the sum of the products of its zeroes taken two at a time, and the product of its zeroes, we can use Vieta's formulas.

Let the zeroes of the cubic polynomial be \( \alpha, \beta, \gamma \). According to Vieta's formulas, we have:

1. The sum of the zeroes: \( \alpha + \beta + \gamma = -a \)
2. The sum of the products of the zeroes taken two at a time: \( \alpha\beta + \beta\gamma + \gamma\alpha = b \)
3. The product of the zeroes: \( \alpha\beta\gamma = -c \)

Given:
- The sum of the zeroes \( \alpha + \beta + \gamma = 2 \)
- The sum of the products of the zeroes taken two at a time \( \alpha\beta + \beta\gamma + \gamma\alpha = -7 \)
- The product of the zeroes \( \alpha\beta\gamma = -14 \)

Now, we can set up our cubic polynomial in the standard form:
\[ p(x) = x^3 + ax^2 + bx + c \]

### Step 1: Determine the coefficients using Vieta's formulas

From the given information, we can express the coefficients \( a, b, c \):
- From \( \alpha + \beta + \gamma = 2 \), we have \( -a = 2 \) or \( a = -2 \).
- From \( \alpha\beta + \beta\gamma + \gamma\alpha = -7 \), we have \( b = -7 \).
- From \( \alpha\beta\gamma = -14 \), we have \( -c = -14 \) or \( c = 14 \).

### Step 2: Write the polynomial

Now substituting the values of \( a, b, c \) into the polynomial:
\[ p(x) = x^3 - 2x^2 - 7x + 14 \]

### Final Answer

Thus, the cubic polynomial is:
\[ \boxed{p(x) = x^3 - 2x^2 - 7x + 14} \]