The roots of the equation `x^(4)-2x^(3)+x-380=0` are

Options

A. `5,-4,(1+-5sqrt(3i))/(2)`
B. `-5,4,(-1+-sqrt(3i))/(2)`
C. `5,4(-1+-sqrt(3i))/(2)`
D. `-5,-4,(1+-5sqrt(3))/(2)`

Explanation

To find the roots of the equation \( x^4 - 2x^3 + x - 380 = 0 \), we will follow these steps:

### Step 1: Identify Possible Rational Roots
Using the Rational Root Theorem, we can test possible rational roots. We will start by testing integer values.

### Step 2: Test \( x = -4 \)
Substituting \( x = -4 \) into the equation:
\[ f(-4) = (-4)^4 - 2(-4)^3 + (-4) - 380 \]
Calculating each term:
\[ = 256 + 64 - 4 - 380 = 256 + 64 - 4 - 380 = 0 \]
Thus, \( x = -4 \) is a root.

### Step 3: Test \( x = 5 \)
Now, let's test \( x = 5 \):
\[ f(5) = (5)^4 - 2(5)^3 + (5) - 380 \]
Calculating each term:
\[ = 625 - 250 + 5 - 380 = 625 - 250 + 5 - 380 = 0 \]
Thus, \( x = 5 \) is also a root.

### Step 4: Factor the Polynomial
Since we have found two roots, \( x = -4 \) and \( x = 5 \), we can express the polynomial as:
\[ (x + 4)(x - 5) \]
Now we need to find the remaining factor by dividing the polynomial by \( (x + 4)(x - 5) \).

### Step 5: Polynomial Long Division
We will divide \( x^4 - 2x^3 + x - 380 \) by \( (x^2 - x - 20) \) (the product of the factors we found):
1. Divide the leading term: \( x^4 \div x^2 = x^2 \)
2. Multiply and subtract:
\[ x^4 - 2x^3 + x - 380 - (x^2)(x^2 - x - 20) = -x^3 + 20x^2 + x - 380 \]
3. Repeat the process until we reach a quadratic.

### Step 6: Solve the Quadratic Equation
Now we have:
\[ x^2 - x + 19 = 0 \]
Using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{1 \pm \sqrt{1 - 76}}{2} = \frac{1 \pm \sqrt{-75}}{2} \]
This gives us:
\[ x = \frac{1 \pm 5i\sqrt{3}}{2} \]

### Final Roots
The roots of the equation \( x^4 - 2x^3 + x - 380 = 0 \) are:
1. \( x = -4 \)
2. \( x = 5 \)
3. \( x = \frac{1 + 5i\sqrt{3}}{2} \)
4. \( x = \frac{1 - 5i\sqrt{3}}{2} \)