Fundamental Theorem of Algebra
Key Questions

Answer:
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Explanation:
Fundamental Theorem of Algebra
The Fundamental Theorem of Algebra (FTOA) tells us that any nonzero polynomial in one variable with complex (possibly real) coefficients has a complex zero.
A straightforward corollary, often stated as part of the FTOA is that a polynomial in a single variable of degree
#n > 0# with complex (possibly real) coefficients has exactly#n# complex (possibly real) zeros, counting multiplicity.To see that the corollary follows, note that if
#f(x)# is a polynomial of degree#n > 0# and#f(a) = 0# , then#(xa)# is a factor of#f(x)# and#f(x)/(xa)# is a polynomial of degree#n1# . So repeatedly applying the FTOA, we find that#f(x)# has exactly#n# complex zeros counting multiplicity.#color(white)()#
DiscriminantsIf you want to know how many real roots a polynomial with real coefficients has, then you might like to look at the discriminant  especially if the polynomial is a quadratic or cubic. Ths discriminant gives less information for polynomials of higher degree.
The discriminant of a quadratic
#ax^2+bx+c# is given by the formula:#Delta = b^24ac# Then:
#Delta > 0# indicates that the quadratic has two distinct real zeros.#Delta = 0# indicates that the quadratic has one real zero of multiplicity two (i.e. a repeated zero).#Delta < 0# indicates that the quadratic has no real zeros. It has a complex conjugate pair of nonreal zeros.The discriminant of a cubic
#ax^3+bx^2+cx+d# is given by the formula:#Delta = b^2c^24ac^34b^3d27a^2d^2+18abcd# Then:
#Delta > 0# indicates that the cubic has three distinct real zeros.#Delta = 0# indicates that the cubic has either one real zero of multiplicity#3# or one real zero of multiplicity#2# and another real zero.#Delta < 0# indicates that the cubic has one real zero and a complex conjugate pair of nonreal zeros.