Polynomial Functions: Extrema, Zeros and Multiplicity

Counting what a degree allows

A polynomial of degree $n$ has at most $n$ real zeros and at most $n-1$ local extrema. Between two consecutive distinct real zeros there is at least one local extremum. If $n$ is even, the function is guaranteed a global maximum or a global minimum (depending on the sign of the leading coefficient); if $n$ is odd, it has neither.

Multiplicity controls the touch

If $(x-c)^m$ is a factor and $m$ is the largest such power, then $c$ is a zero of multiplicity $m$:

• $m$ odd $\Rightarrow$ the graph crosses the $x$-axis at $c$ (and flattens as it crosses when $m\ge3$);
• $m$ even $\Rightarrow$ the graph is tangent to the $x$-axis at $c$: it touches and turns back without changing sign.

Complex zeros complete the count

Counted with multiplicity, a degree-$n$ polynomial has exactly $n$ complex zeros. When the coefficients are real, non-real zeros arrive in conjugate pairs $a\pm bi$ — so the number of non-real zeros is always even, and an odd-degree polynomial with real coefficients must have at least one real zero.