Key Concepts & Glossary
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 have opposite signs, then there exists at least one value c between a and b for which .
Glossary
- global maximum
- highest turning point on a graph; where for all x.
- global minimum
- lowest turning point on a graph; where for all x.
- Intermediate Value Theorem
- for two numbers a and b in the domain of f, if and , then the function f takes on every value between and ; 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 , is a zero of multiplicity p.