Question

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

Answer 1

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 `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 `(x-a)` is a factor of `f(x)` and `f(x)/(x-a)` is a polynomial of degree `n-1`. So repeatedly applying the FTOA, we find that `f(x)` has exactly `n` complex zeros counting multiplicity.

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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 `ax^2+bx+c` is given by the formula:

`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 `ax^3+bx^2+cx+d` is given by the formula:

`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.