Form the polynomial equation of degree 3 whose roots are 2,3 and 6.

Answer

To form a polynomial equation of degree 3 with roots 2, 3, and 6, we can follow these steps:

### Step 1: Identify the roots
The roots of the polynomial are given as:
- \( \alpha = 2 \)
- \( \beta = 3 \)
- \( \gamma = 6 \)

### Step 2: Write the polynomial in factored form
The polynomial can be expressed in factored form as:
\[ P(x) = (x - \alpha)(x - \beta)(x - \gamma) \]
Substituting the values of the roots:
\[ P(x) = (x - 2)(x - 3)(x - 6) \]

### Step 3: Expand the polynomial
First, we will expand the first two factors:
\[ (x - 2)(x - 3) = x^2 - 3x - 2x + 6 = x^2 - 5x + 6 \]
Now, we will multiply this result by the third factor \( (x - 6) \):
\[ P(x) = (x^2 - 5x + 6)(x - 6) \]

### Step 4: Distribute the terms
Now, we will distribute \( (x - 6) \) across \( (x^2 - 5x + 6) \):
\[ P(x) = x^2(x - 6) - 5x(x - 6) + 6(x - 6) \]
Calculating each term:
1. \( x^2(x - 6) = x^3 - 6x^2 \)
2. \( -5x(x - 6) = -5x^2 + 30x \)
3. \( 6(x - 6) = 6x - 36 \)

### Step 5: Combine like terms
Now, we combine all the terms:
\[ P(x) = x^3 - 6x^2 - 5x^2 + 30x + 6x - 36 \]
Combining like terms:
\[ P(x) = x^3 - 11x^2 + 36x - 36 \]

### Step 6: Write the final polynomial equation
Thus, the polynomial equation of degree 3 whose roots are 2, 3, and 6 is:
\[ x^3 - 11x^2 + 36x - 36 = 0 \]