Section 3.2 - Polynomials: Graphs and Properties

Section Objectives

  1. Find the zeros of a polynomial and determine their multiplicities.
  2. Determine the end behavior of a polynomial function.
  3. Use intercepts and end behavior to graph a polynomial function.



Polynomials

An -th-degree polynomial in the variable is a function of the form

,

where the coefficients , , , are real or complex numbers with .



The zeros of a function are those -values for which . The real zeros of correspond to the -intercepts of the graph of : if the number is a zero of , then is an -intercept of the graph.




To find the zeros of a polynomial function, we must solve a polynomial equation of the form



General polynomial equations are often solved by first making one side of the equation equal to zero. After doing so, the next steps may vary depending on the nature of the polynomial. For now, we will focus on solving by factoring.



Factored polynomials and zeros

When a polynomial is completely factored, the number of times a specific linear factor occurs in the factorization is called the multiplicity of the corresponding zero.




Examples










End behavior, zeros, and polynomial graphs

The general shape of the graph of a polynomial can be easily determined from the polynomial's factored form.




Example