# How do you know how many roots are in a polynomial?

##### 1 Answer

See explanation...

#### Explanation:

**Fundamental Theorem of Algebra**

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

**Discriminants**

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#

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 non-real zeros.

The discriminant of a cubic

#Delta = b^2c^2-4ac^3-4b^3d-27a^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 non-real zeros.