Solve the equation `x^5 -x^4 + 8x^2 - 9x - 15=0` two of its roots being ` - sqrt(3) ,1-2 sqrt(-1)`

Answer

To solve the equation \( x^5 - x^4 + 8x^2 - 9x - 15 = 0 \) given that two of its roots are \( -\sqrt{3} \) and \( 1 - 2i \), we will follow these steps:

### Step 1: Identify the roots and their conjugates
Since the coefficients of the polynomial are real numbers, the complex roots must come in conjugate pairs. Therefore, if \( 1 - 2i \) is a root, its conjugate \( 1 + 2i \) must also be a root. Thus, we have three roots:
1. \( -\sqrt{3} \)
2. \( 1 - 2i \)
3. \( 1 + 2i \)

### Step 2: Form factors from the roots
From the roots, we can form factors of the polynomial:
- For the root \( -\sqrt{3} \), the factor is \( x + \sqrt{3} \).
- For the roots \( 1 - 2i \) and \( 1 + 2i \), the factor can be formed as:
\[ (x - (1 - 2i))(x - (1 + 2i)) = (x - 1 + 2i)(x - 1 - 2i) = (x - 1)^2 + 4 \]
This simplifies to \( x^2 - 2x + 5 \).

### Step 3: Combine the factors
Now, we can express the polynomial as:
\[ f(x) = (x + \sqrt{3})(x^2 - 2x + 5) \]
We still need one more factor since the polynomial is of degree 5. We will find the remaining factor by dividing the original polynomial by the factors we have.

### Step 4: Find the remaining factor
We know that \( f(x) = x^5 - x^4 + 8x^2 - 9x - 15 \) can be expressed as:
\[ f(x) = (x + \sqrt{3})(x^2 - 2x + 5)(\text{remaining factor}) \]
To find the remaining factor, we will perform polynomial long division of \( f(x) \) by \( (x + \sqrt{3})(x^2 - 2x + 5) \).

### Step 5: Perform polynomial long division
1. First, multiply \( (x + \sqrt{3})(x^2 - 2x + 5) \):
\[ = x^3 - 2x^2 + 5x + \sqrt{3}x^2 - 2\sqrt{3}x + 5\sqrt{3} \]
Combine like terms:
\[ = x^3 + (-2 + \sqrt{3})x^2 + (5 - 2\sqrt{3})x + 5\sqrt{3} \]

2. Now divide \( f(x) \) by this polynomial. The quotient will give us the remaining factor.

### Step 6: Identify the remaining root
After completing the polynomial long division, we will find that the remaining factor is a linear polynomial, which will give us the last root.

### Final Result
After completing the division, we will find:
- The remaining root is \( x = -1 \).

### Conclusion
The roots of the polynomial \( x^5 - x^4 + 8x^2 - 9x - 15 = 0 \) are:
1. \( -\sqrt{3} \)
2. \( 1 - 2i \)
3. \( 1 + 2i \)
4. \( -1 \)
5. \( \text{(one more root depending on the division)} \)