list all solutions if x^(4)+x^(3)-11x^(2)+9x+20=0 and if one of the roots is x=2-i
To find the solutions to the given equation, we will make use of a method called the polynomial factoring or the Factor theorem. The key idea is that if `a` is a root of a polynomial equation, then `(x - a)` will be a factor of the polynomial.
Given that `x = 2 - i` is a root, we know that `(x - (2 - i)) = (x - 2 + i)` will be a factor of the polynomial. We can use this factor to find additional roots.
Now, let's simplify the factor `(x - 2 + i)` by finding its conjugate. The conjugate of `2 - i` is `2 + i`. Multiplying the two conjugates, we get:
`(x - 2 + i)(x - 2 - i) = (x - 2)^2 - (i)^2 = (x - 2)^2 + 1`
Therefore, the simplified factor is `(x - 2)^2 + 1`.
To find the other factors, we can divide the original polynomial by `(x - 2)^2 + 1`. Performing polynomial division, we have:
```
x^2 + 3x - 4
----------------------
(x^2 - 4x + 5)(x - 2)^2 + 1
```
Since the degree of `(x - 2)^2 + 1` is 2, we expect to get a quadratic polynomial as the quotient. In this case, the quotient is `x^2 + 3x - 4`.
Now, we need to find the roots of the quadratic polynomial `x^2 + 3x - 4`. We can solve it by factoring or using the quadratic formula.
Factoring the quadratic polynomial, we have:
`(x + 4)(x - 1) = 0`
Setting each factor to zero, we get:
x + 4 = 0 -> x = -4
x - 1 = 0 -> x = 1
Therefore, the solutions to the equation x^4 + x^3 - 11x^2 + 9x + 20 = 0, given that x = 2 - i is a root, are:
x = 2 - i, x = 2 + i, x = -4, and x = 1.