Why is it remarkable that with real coefficients, there must be complex solutions?

it is important to learn about complex solution because polynomial equation formed over a complex number can only be solved by a complex number.

this is so because, the fundamental theorem of algebra state that every polynomial equation in one variable with complex coefficient has at least one complex solution.

Step-by-step explanation:

it is important to learn about complex solution because polynomial equation formed over a complex number can only be solved by a complex number.

this is so because, the fundamental theorem of algebra state that every polynomial equation in one variable with complex coefficient has at least one complex solution.

for example:

Given any positive integer n ≥ 1 and any choice of complex numbers a0,a1,...,an, such that an 6= 0,

the polynomial equation

anzn +···+ a1z + a0 = 0 (1) has at least one solution z ∈C.

No analogous result holds for guaranteeing that a real solution exists to Equation (1) if we restrict the coefficients a0,a1,...,an to be real numbers.

E.g., there does not exist a real number x satisfying an equation as simple as x2 + 1 = 0. Similarly, the consideration of polynomial equations having integer (resp. rational) coefficients quickly forces us to consider solutions that cannot possibly be integers (resp. rational numbers).