# What is the maximum number of real roots a polynomial of any degree can have?

Is the following reasoning correct?

According to the complex conjugate root theorem, the number of complex roots of a polynomial is always equal to its degree.

Since odd degree polynomials have a maximum of 2 turning points, they can have a maximum of 3 real roots. And since even degree polynomials have a maximum of 1 turning point, they can have a maximum of 3 real roots.

Therefore, the maximum number of a real roots a polynomial of any degree can have is 3, all other roots are non-real.

• $(x-1)(x-2)\dots(x-n)$ has $n$ distinct real roots. A polynomial of degree $n$ has at most $n-1$ turning points. Commented Jul 9, 2017 at 13:51
• onemathematicalcat.org/Math/Precalculus_obj/turningPoints.htm
– user9464
Commented Jul 9, 2017 at 14:05

Actually the maximum number of turning points of a degree $n$ polynomial is $n-1$ and the one in the picture indeed has four of them, and five real roots.
Generalizing, if $n$ is any positive integer, the polynomial $$(x-1)(x-2)\dots(x-n)$$ has $n$ distinct roots.