The equation whose roots are ` 1+- 2 i,0,1` is

Options

A. `x(x^3 +3x^2 -7x +4)=0`
B. `x(x^3 -3x^2 +7x -5)=0`
C. `x(x^3-3x^2 -7x +4)=0`
D. `x(x^3 -3x^2 +7x -4)=0`

Answer

To find the equation whose roots are \(1 \pm 2i\), \(0\), and \(1\), we can follow these steps:

### Step 1: Identify the roots
The roots of the equation are given as:
- \(1 + 2i\)
- \(1 - 2i\)
- \(0\)
- \(1\)

### Step 2: Write the factors corresponding to the roots
The factors of the polynomial corresponding to these roots will be:
- For the root \(1 + 2i\): \(x - (1 + 2i) = x - 1 - 2i\)
- For the root \(1 - 2i\): \(x - (1 - 2i) = x - 1 + 2i\)
- For the root \(0\): \(x - 0 = x\)
- For the root \(1\): \(x - 1\)

Thus, the polynomial can be expressed as:
\[ (x - (1 + 2i))(x - (1 - 2i))(x)(x - 1) \]

### Step 3: Simplify the complex factors
First, we will simplify the product of the complex conjugate factors:
\[ (x - 1 - 2i)(x - 1 + 2i) \]
Using the difference of squares:
\[ = (x - 1)^2 - (2i)^2 = (x - 1)^2 - 4(-1) = (x - 1)^2 + 4 \]

### Step 4: Expand \((x - 1)^2 + 4\)
Now, we expand \((x - 1)^2 + 4\):
\[ = (x^2 - 2x + 1) + 4 = x^2 - 2x + 5 \]

### Step 5: Multiply by the remaining factors
Now we multiply this result by the remaining factors:
\[ (x^2 - 2x + 5)(x)(x - 1) \]

First, we multiply \((x^2 - 2x + 5)\) by \(x\):
\[ x(x^2 - 2x + 5) = x^3 - 2x^2 + 5x \]

Now, we multiply this result by \((x - 1)\):
\[ (x^3 - 2x^2 + 5x)(x - 1) \]
Expanding this gives:
\[ = x^4 - x^3 - 2x^3 + 2x^2 + 5x^2 - 5x \]
Combining like terms:
\[ = x^4 - 3x^3 + 7x^2 - 5x \]

### Step 6: Write the final equation
Thus, the equation whose roots are \(1 \pm 2i\), \(0\), and \(1\) is:
\[ x^4 - 3x^3 + 7x^2 - 5x = 0 \]