Von der Gleichung \(x^{4}-2 x^{3}+x^{2}+2 x-2=0\) ist eine (komplexe) L?sung bekannt: \(x_{1}=1-\mathrm{j}\). Wie lauten die übrigen Lösungen?

When dealing with complex numbers, the complex conjugate is an important concept. The complex conjugate of a complex number is formed by changing the sign of the imaginary part. For example, if the complex number is 1 - j, its complex conjugate would be 1 + j.
Complex conjugates always come in pairs. If a polynomial has complex coefficients, its roots must include both the complex number and its conjugate. This property helps us to simplify and solve polynomial equations effectively by ensuring all roots are accounted for.

Polynomial division is used to simplify complex polynomial equations. It's akin to long division but applied to polynomials. In our problem, after factoring out the quadratic polynomial, we need to divide the original polynomial by this factor.
Let's say we need to divide P(x) by a quadratic polynomial Q(x). The result will give us another polynomial, often simpler than the original. This result helps break down complex polynomials into more manageable parts, making it easier to find all roots.
Both long polynomial division and synthetic division can be used for this task. Choose the method you find easier or more appropriate for the given equation.

Quadratic equations are polynomials of degree two and have the general form ax^2 + bx + c = 0. They play a crucial role in solving higher-degree polynomials because they can often be factored out of these polynomials.
In our example, once we've identified that (x - (1 - j))(x - (1 + j)) = x^2 - 2x + 2 is a factor of the original fourth-degree polynomial, we can divide by this quadratic to get the remaining factor. The final step involves solving these simpler quadratic equations to find the remaining roots of the polynomial.

Complex numbers are a combination of real and imaginary parts, generally written as a + bi, where a is the real part and b is the imaginary part. In polynomial equations, roots can be complex numbers, especially when dealing with higher-degree polynomials.
Understanding how to manipulate and work with complex numbers is crucial. This includes knowing operations like addition, subtraction, and finding the complex conjugate. In our problem, recognizing that 1 - j is a root and immediately knowing its conjugate 1 + j is part of solving the polynomial efficiently.
By mastering complex numbers, you can tackle a wide variety of mathematical problems, providing a deeper understanding of polynomial equations and their roots.