From the polynomial equation whose roots are 3,2,1+i,1-i

Answer

To form a polynomial equation whose roots are given as 3, 2, 1+i, and 1-i, we can follow these steps:

### Step 1: Identify the Roots
The roots of the polynomial are:
- \( \alpha = 3 \)
- \( \beta = 2 \)
- \( \gamma = 1 + i \)
- \( \delta = 1 - i \)

### Step 2: Write the Polynomial in Factored Form
The polynomial can be expressed in factored form as:
\[ P(x) = (x - \alpha)(x - \beta)(x - \gamma)(x - \delta) \]
Substituting the roots:
\[ P(x) = (x - 3)(x - 2)(x - (1 + i))(x - (1 - i)) \]

### Step 3: Simplify the Complex Roots
First, simplify the product of the complex roots:
\[ (x - (1 + i))(x - (1 - i)) = (x - 1 - i)(x - 1 + i) \]
This can be recognized as a difference of squares:
\[ = (x - 1)^2 - (i)^2 \]
Since \( i^2 = -1 \):
\[ = (x - 1)^2 - (-1) = (x - 1)^2 + 1 \]
Now, expanding \( (x - 1)^2 \):
\[ = x^2 - 2x + 1 + 1 = x^2 - 2x + 2 \]

### Step 4: Combine All Factors
Now substitute back into the polynomial:
\[ P(x) = (x - 3)(x - 2)(x^2 - 2x + 2) \]

### Step 5: Multiply the First Two Factors
First, multiply \( (x - 3)(x - 2) \):
\[ = x^2 - 5x + 6 \]

### Step 6: Multiply the Result with the Quadratic
Now multiply \( (x^2 - 5x + 6) \) with \( (x^2 - 2x + 2) \):
\[ P(x) = (x^2 - 5x + 6)(x^2 - 2x + 2) \]

### Step 7: Expand the Polynomial
Expanding this product:
1. Multiply \( x^2 \) with each term in the second polynomial:
\[ x^2 \cdot x^2 = x^4 \]
\[ x^2 \cdot (-2x) = -2x^3 \]
\[ x^2 \cdot 2 = 2x^2 \]

2. Multiply \( -5x \) with each term in the second polynomial:
\[ -5x \cdot x^2 = -5x^3 \]
\[ -5x \cdot (-2x) = 10x^2 \]
\[ -5x \cdot 2 = -10x \]

3. Multiply \( 6 \) with each term in the second polynomial:
\[ 6 \cdot x^2 = 6x^2 \]
\[ 6 \cdot (-2x) = -12x \]
\[ 6 \cdot 2 = 12 \]

### Step 8: Combine Like Terms
Now combine all the terms:
\[ P(x) = x^4 + (-2x^3 - 5x^3) + (2x^2 + 10x^2 + 6x^2) + (-10x - 12x) + 12 \]
This simplifies to:
\[ P(x) = x^4 - 7x^3 + 18x^2 - 22x + 12 \]

### Final Polynomial
The polynomial equation whose roots are 3, 2, 1+i, and 1-i is:
\[ x^4 - 7x^3 + 18x^2 - 22x + 12 = 0 \]

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