Find the roots of `x^4-16x^3+86x^2-176x+105=0`

Answer

To find the roots of the polynomial equation \( x^4 - 16x^3 + 86x^2 - 176x + 105 = 0 \), we will follow these steps:

### Step 1: Define the Polynomial
Let \( f(x) = x^4 - 16x^3 + 86x^2 - 176x + 105 \).

### Step 2: Check for Rational Roots
We will use the Rational Root Theorem to test possible rational roots. Let's start by testing \( x = 1 \).

Calculating \( f(1) \):
\[ f(1) = 1^4 - 16(1^3) + 86(1^2) - 176(1) + 105 \]
\[ = 1 - 16 + 86 - 176 + 105 = 0 \]

Since \( f(1) = 0 \), \( x = 1 \) is a root of the polynomial.

### Step 3: Polynomial Division
Now that we have found one root, we can perform synthetic division of \( f(x) \) by \( x - 1 \).

Using synthetic division:
- Coefficients of \( f(x) \): \( 1, -16, 86, -176, 105 \)
- Divide by \( 1 \):

\[ \begin{array}{r|rrrrr} 1 & 1 & -16 & 86 & -176 & 105 \\ & & 1 & -15 & 71 & -105 \\ \hline & 1 & -15 & 71 & -105 & 0 \\ \end{array} \]

The result of the synthetic division is \( x^3 - 15x^2 + 71x - 105 \).

### Step 4: Find Roots of the Cubic Polynomial
Let \( g(x) = x^3 - 15x^2 + 71x - 105 \).

We will check for rational roots again. Let's test \( x = 3 \).

Calculating \( g(3) \):
\[ g(3) = 3^3 - 15(3^2) + 71(3) - 105 \]
\[ = 27 - 135 + 213 - 105 = 0 \]

Since \( g(3) = 0 \), \( x = 3 \) is also a root.

### Step 5: Synthetic Division Again
Now we will perform synthetic division of \( g(x) \) by \( x - 3 \).

Using synthetic division:
- Coefficients of \( g(x) \): \( 1, -15, 71, -105 \)
- Divide by \( 3 \):

\[ \begin{array}{r|rrrr} 3 & 1 & -15 & 71 & -105 \\ & & 3 & -36 & 105 \\ \hline & 1 & -12 & 35 & 0 \\ \end{array} \]

The result is \( x^2 - 12x + 35 \).

### Step 6: Factor the Quadratic Polynomial
Now we need to factor \( x^2 - 12x + 35 \).

This can be factored as:
\[ x^2 - 12x + 35 = (x - 5)(x - 7) \]

### Step 7: Combine All Roots
Now we have all the roots:
- From \( f(x) \): \( x - 1 = 0 \) gives \( x = 1 \)
- From \( g(x) \): \( x - 3 = 0 \) gives \( x = 3 \)
- From \( x^2 - 12x + 35 = 0 \): gives \( x = 5 \) and \( x = 7 \)

### Final Roots
Thus, the roots of the polynomial \( f(x) = 0 \) are:
\[ \boxed{1, 3, 5, 7} \]