##1+z+z^2+\dots +z^{n-1} = \frac{z^n - 1}{z-1}##. I.e. the roots of ##z^n-1## are ##1## plus the roots of ##1+z+z^2+\dots +z^{n-1}##. They are called roots of unity (##z^n = 1 ##) and are equally distributed along the unit circle, starting at ##(1,0)##. Therefore they are multiples of the angle ##\frac{2 \pi}{n}##, i.e. ##\exp (i{\frac{k}{n} 2 \pi}) ## with ##k=0,\dots ,n-1##.TheCanadian said:
Factoring a complex polynomial means breaking down a polynomial with complex coefficients into simpler terms. This is done by finding the roots or solutions of the polynomial, which are the values of the variables that make the polynomial equal to zero.
Factoring a complex polynomial is important because it helps us solve equations, find intercepts, and simplify expressions. It also allows us to understand the behavior and properties of a polynomial better.
The steps involved in factoring a complex polynomial include grouping, factoring out the greatest common factor, and using techniques such as the quadratic formula, completing the square, or the difference of squares method to factor the remaining terms.
Some common techniques used in factoring a complex polynomial include the quadratic formula, completing the square, and the difference of squares method. Other techniques include factoring by grouping, factoring by substitution, and factoring by trial and error.
A complex polynomial can be factored if it has at least two terms and each term has a variable with an exponent greater than or equal to 1. If the polynomial cannot be factored, it is called a prime polynomial. In some cases, a polynomial can be factored further using more advanced techniques such as the rational roots theorem or the sum/difference of cubes formula.
