How do you know how many roots are in a polynomial?
The Fundamental Theorem of Algebra (FTOA) tells us that any non-zero 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
To see that the corollary follows, note that if
If 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
#Delta = b^2-4ac#
#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 non-real zeros.
The discriminant of a cubic
#Delta = b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd#
#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 non-real zeros.