Solve the following equations
`x^5 -5x^4 +9x^3 -9x^2 +5x -1=0`

Answer

To solve the equation \( x^5 - 5x^4 + 9x^3 - 9x^2 + 5x - 1 = 0 \), we will follow these steps:

### Step 1: Identify the Polynomial
Let \( f(x) = x^5 - 5x^4 + 9x^3 - 9x^2 + 5x - 1 \).

### Step 2: Use the Rational Root Theorem
We will test for possible rational roots using the Rational Root Theorem. A good starting point is to test \( x = 1 \).

### Step 3: Evaluate \( f(1) \)
Calculate \( f(1) \):
\[ f(1) = 1^5 - 5(1^4) + 9(1^3) - 9(1^2) + 5(1) - 1 \]
\[ = 1 - 5 + 9 - 9 + 5 - 1 = 0 \]
Since \( f(1) = 0 \), \( x = 1 \) is a root of the polynomial.

### Step 4: Factor the Polynomial
Since \( x = 1 \) is a root, we can factor \( f(x) \) as:
\[ f(x) = (x - 1)(\text{quotient}) \]
We will perform polynomial long division of \( f(x) \) by \( (x - 1) \).

### Step 5: Perform Polynomial Long Division
Dividing \( x^5 - 5x^4 + 9x^3 - 9x^2 + 5x - 1 \) by \( x - 1 \):
1. Divide the leading term: \( x^5 \div x = x^4 \).
2. Multiply \( x^4 \) by \( (x - 1) \): \( x^5 - x^4 \).
3. Subtract:
\[ (-5x^4 + x^4) + 9x^3 - 9x^2 + 5x - 1 = -4x^4 + 9x^3 - 9x^2 + 5x - 1 \]
4. Repeat the process:
- Divide \( -4x^4 \div x = -4x^3 \).
- Multiply: \( -4x^4 + 4x^3 \).
- Subtract:
\[ (9x^3 - 4x^3) - 9x^2 + 5x - 1 = 5x^3 - 9x^2 + 5x - 1 \]
5. Continue until reaching the constant term.

After completing the division, we find:
\[ f(x) = (x - 1)(x^4 - 4x^3 + 5x^2 - 4x + 1) \]

### Step 6: Solve the Quartic Polynomial
Now we need to solve \( x^4 - 4x^3 + 5x^2 - 4x + 1 = 0 \).

### Step 7: Use Substitution
Let \( y = x + \frac{1}{x} \). Then we can rewrite the quartic polynomial in terms of \( y \).

### Step 8: Solve for \( y \)
Rearranging gives us a quadratic in \( y \):
\[ y^2 - 4y + 3 = 0 \]
Factoring gives:
\[ (y - 3)(y - 1) = 0 \]
Thus, \( y = 3 \) or \( y = 1 \).

### Step 9: Solve for \( x \)
1. For \( y = 3 \):
\[ x + \frac{1}{x} = 3 \implies x^2 - 3x + 1 = 0 \]
Using the quadratic formula:
\[ x = \frac{3 \pm \sqrt{5}}{2} \]
2. For \( y = 1 \):
\[ x + \frac{1}{x} = 1 \implies x^2 - x + 1 = 0 \]
The roots are:
\[ x = \frac{1 \pm \sqrt{-3}}{2} = \frac{1 \pm \sqrt{3}i}{2} \]

### Step 10: Compile All Roots
The roots of the original equation are:
1. \( x = 1 \)
2. \( x = \frac{3 + \sqrt{5}}{2} \)
3. \( x = \frac{3 - \sqrt{5}}{2} \)
4. \( x = \frac{1 + \sqrt{3}i}{2} \)
5. \( x = \frac{1 - \sqrt{3}i}{2} \)

### Final Answer
The solutions to the equation \( x^5 - 5x^4 + 9x^3 - 9x^2 + 5x - 1 = 0 \) are:
\[ x = 1, \quad x = \frac{3 + \sqrt{5}}{2}, \quad x = \frac{3 - \sqrt{5}}{2}, \quad x = \frac{1 + \sqrt{3}i}{2}, \quad x = \frac{1 - \sqrt{3}i}{2} \]