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 will follow these steps:

### Step 1: Identify the values from the problem
We are given:
- Sum of the zeroes (α + β + γ) = 5
- Sum of the products of the zeroes taken two at a time (αβ + αγ + βγ) = -6
- Product of the zeroes (αβγ) = 12

### Step 2: Write the general form of a cubic polynomial
The general form of a cubic polynomial with roots α, β, and γ is given by:
\[ P(x) = x^3 - (α + β + γ)x^2 + (αβ + αγ + βγ)x - (αβγ) \]

### Step 3: Substitute the values into the polynomial
Now, we will substitute the values we identified in Step 1 into the polynomial:
- \( α + β + γ = 5 \)
- \( αβ + αγ + βγ = -6 \)
- \( αβγ = 12 \)

Substituting these values into the polynomial form:
\[ P(x) = x^3 - (5)x^2 + (-6)x - (12) \]

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

### Step 5: Write the final polynomial
Thus, the cubic polynomial we are looking for is:
\[ P(x) = x^3 - 5x^2 - 6x - 12 \]