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another way to present the fundamental theorem of algebra is to state that, "for any polynomial equation of degree n with real coefficients, there exist n complex solutions." why is it important to learn about complex solutions? 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).

d. vertex

step-by-step explanation:

answer: (-6,3)

step-by-step explanation: