Polynomial Functions

Definition and Key Properties

A polynomial function in one variable is a function of the form:

• Coefficients: The constants are called the coefficients.

• Leading Coefficient: The coefficient of the highest power of is called the leading coefficient.

• Degree: The degree of the polynomial is the highest power of with a nonzero coefficient.

Example 1: Determine which of the following are polynomials. For those that are, state the degree.

• — Not a polynomial (contains negative exponent)

• — Polynomial, degree 4

• — Not a polynomial (contains fractional exponent)

• — Not a polynomial (rational function)

• — Polynomial, degree 0

• — Polynomial, degree 4

Power Functions

A power function is a function of the form , where is a positive integer.

• Domain: All real numbers ()

• Range: Depends on (even : ; odd : )

• Symmetry: Even → even function (symmetric about -axis); odd → odd function (symmetric about origin)

• Key Points: , ,

• End Behavior: For even , both ends go to ; for odd , left end goes to , right end to

Example: The graph shows , , .

Graphing Polynomial Functions

Graphing Using Transformations

Polynomial functions can be graphed by applying transformations such as shifts, stretches, and reflections.

• Vertical and horizontal shifts: shifts up/down; shifts right/left.

• Reflections: reflects across the -axis.

• Stretches/compressions: stretches if , compresses if .

Example 2: Sketch the graphs of the following functions:

• — Parabola opening downward, vertex at

• — Quartic function, vertex at , compressed vertically

Zeros of Polynomial Functions and Their Multiplicity

Identifying Real Zeros and Multiplicity

The real zeros of a polynomial function are the -values where . The multiplicity of a zero refers to how many times a particular zero occurs.

• If , then is a zero of .

• is an -intercept of the graph of .

• is a factor of .

• is a solution to the equation .

Example 3: Find a polynomial function of degree 3 whose real zeros are , $2.

Example 4: Find the real zeros and their multiplicities for the following polynomial:

• Zero at , multiplicity 2

• Zero at , multiplicity 1

• Zero at , multiplicity 4

Table: Evaluating a Polynomial at Given Points

Example 5: Consider the polynomial . Complete the following table of values:

Use these values to sketch a possible graph of .

• If is a real zero of even multiplicity, the graph touches the -axis at and turns around.

• If is a real zero of odd multiplicity, the graph crosses the -axis at .

Degree and End Behavior of Polynomial Functions

Degree and Graph Shape

The degree of a polynomial determines the maximum number of real zeros and turning points the graph can have.

• A polynomial of degree can have at most real zeros.

• The graph can have at most turning points.

Example 6: Given a graph, determine the least possible degree the polynomial can have by counting the number of turning points and zeros.

End Behavior

Fact: The end behavior of the graph of the polynomial function

is the same as that of the graph of the power function

• If is even and , both ends go to .

• If is even and , both ends go to .

• If is odd and , left end goes to , right end to .

• If is odd and , left end goes to , right end to .

Matching Graphs to Polynomial Equations

Example 7: Which of the following could be the graph of ?

• Analyze the degree (4), leading coefficient (positive), and number of turning points to match the graph.

Additional info: These notes cover the main concepts of polynomial functions, including definitions, graphing techniques, zeros and their multiplicities, degree, and end behavior, which are essential topics in College Algebra.