4.3: Polynomial Functions

Types of Polynomial Functions I. Polynomial Functions Types of Polynomial Functions Degree Name Constant 1 Linear 2 Quadratic 3 Cubic 4 Quartic 5 Quintic

Typical polynomial graphs Cubic graphs Quartic graphs

lim 𝑥→∞ 𝑓(𝑥) lim 𝑥→−∞ 𝑓(𝑥) −∞ ∞ ∞ −∞ ∞ ∞ −∞ −∞ End behavior of polynomial functions. Example 3: List the end behavior of the following functions lim 𝑥→∞ 𝑓(𝑥) Function lim 𝑥→−∞ 𝑓(𝑥) −∞ ∞ ∞ −∞ ∞ ∞ −∞ −∞

Zeros of Polynomial Functions Example 4: Find the zeroes of the function below.

B. Repeated Zeros: when a factor is repeated, it’s zero is called a repeated zero. C. Definition: If f is a polynomial function and 𝑥−𝑐 𝑚 is a factor of f, then c is a zero of multiplicity m of f.

Example 5: Find the multiplicity of each of the zeros for the function below, then sketch a graph of the function.

Zeros of odd and even multiplicity. If a polynomial function f has a real zero c of odd multiplicity, then the graph of f crosses the x-axis at (c,0) and the value of f changes sign at x = c. If a polynomial function f has a real zero c of even multiplicity, then the graph of f does not cross the x-axis at (c,0) and the value of f does not changes sign at x = c. 𝑓 𝑥 = 𝑥+3 4 𝑥−2 3 𝐸𝑣𝑒𝑛 𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦 𝑂𝑑𝑑 𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦

Example 6: sketching the graph of a factored polynomial.