Find the cubic polynomial with the sum, sum of the products of its zeroes taken two at a time and the product of its zeroes are 5, -6, 12 respectively.

Explanation

To find the 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 follow these steps:

### Step 1: Identify the roots and their relationships
Let the roots of the cubic polynomial be \( \alpha, \beta, \gamma \).
From the problem, we know:
- The sum of the roots: \( \alpha + \beta + \gamma = 5 \)
- The sum of the products of the roots taken two at a time: \( \alpha\beta + \beta\gamma + \gamma\alpha = -6 \)
- The product of the roots: \( \alpha\beta\gamma = 12 \)

### Step 2: Write the general form of the cubic polynomial
The general form of a cubic polynomial with roots \( \alpha, \beta, \gamma \) can be expressed as:
\[ P(x) = x^3 - (\alpha + \beta + \gamma)x^2 + (\alpha\beta + \beta\gamma + \gamma\alpha)x - \alpha\beta\gamma \]

### Step 3: Substitute the known values into the polynomial
Now, we substitute the values we found in Step 1 into the polynomial:
\[ P(x) = x^3 - (5)x^2 + (-6)x - (12) \]

### Step 4: Simplify the polynomial
Now we simplify the polynomial:
\[ P(x) = x^3 - 5x^2 - 6x - 12 \]

### Final Answer
Thus, the cubic polynomial is:
\[ \boxed{x^3 - 5x^2 - 6x - 12} \]