Polynomials of degree n are linear combination of powers of a variable, i.e. they are the sum of all the powers of the variable from 0 to n, with some coefficients. So, a generic polynomial of degree n is written as
[tex] a_nx^n+a_{n-1}x^{n-1}+...+a_2x^2+a_1x+a_0 [/tex]
The fundamental theorem of algebra states that a polynomial of degree n has exactly n roots.
Moreover, polynomials with real coefficients also have the following property: if [tex] \alpha \in \mathbb{C} [/tex] is a solution of a polynomial, its conjugate [tex] \overline{\alpha} [/tex] is also a solution of the same polynomial.
The conjugate is obtained by changing the sign of the complex part of the number:
[tex] \alpha = a+bi \to \overline{\alpha} = a-bi [/tex]
So, if [tex] -3+i [/tex] is a solution of the polynomial also its conjugate [tex] -3-i [/tex] is a solution of the polynomial
