## Key Concepts

• Polynomial functions of degree 2 or more are smooth, continuous functions.
• To find the zeros of a polynomial function, if it can be factored, factor the function and set each factor equal to zero.
• Another way to find the x-intercepts of a polynomial function is to graph the function and identify the points at which the graph crosses the x-axis.
• The multiplicity of a zero determines how the graph behaves at the x-intercepts.
• The graph of a polynomial will cross the horizontal axis at a zero with odd multiplicity.
• The graph of a polynomial will touch the horizontal axis at a zero with even multiplicity.
• The end behavior of a polynomial function depends on the leading term.
• The graph of a polynomial function changes direction at its turning points.
• A polynomial function of degree n has at most n – 1 turning points.
• To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has at most n – 1 turning points.
• Graphing a polynomial function helps to estimate local and global extremas.
• The Intermediate Value Theorem tells us that if $f\left(a\right) \text{and} f\left(b\right)$ have opposite signs, then there exists at least one value c between a and b for which $f\left(c\right)=0$.

## Glossary

global maximum
highest turning point on a graph; $f\left(a\right)$ where $f\left(a\right)\ge f\left(x\right)$ for all x.
global minimum
lowest turning point on a graph; $f\left(a\right)$ where $f\left(a\right)\le f\left(x\right)$
for all x.
Intermediate Value Theorem
for two numbers a and b in the domain of f, if $a<b$ and $f\left(a\right)\ne f\left(b\right)$, then the function f takes on every value between $f\left(a\right)$ and $f\left(b\right)$; specifically, when a polynomial function changes from a negative value to a positive value, the function must cross the x-axis
multiplicity
the number of times a given factor appears in the factored form of the equation of a polynomial; if a polynomial contains a factor of the form ${\left(x-h\right)}^{p}$, $x=h$ is a zero of multiplicity p.