Identify the x-Intercepts of Polynomial Functions whose Equations are Factorable

This section will dig deeper into the relationship between the graph of a polynomial function and its equation. You will see how to use the factors of a polynomial function to determine where the x-intercepts are, and you will also learn about the multiplicity of a zero (x-intercept) and how to find it.

Learning Objectives

In this section, you will:

  • Recognize characteristics of graphs of polynomial functions.
  • Use factoring to find zeros of polynomial functions.
  • Identify zeros and their multiplicities.
  • Determine end behavior.
  • Understand the relationship between degree and turning points.
  • Graph polynomial functions.
  • Use the Intermediate Value Theorem.

The revenue in millions of dollars for a fictional cable company from 2006 through 2013 is shown in Table 1.

Year 2006 2007 2008 2009 2010 2011 2012 2013
Revenues 52.4 52.8 51.2 49.5 48.6 48.6 48.7 47.1

Table 1


The revenue can be modeled by the polynomial function

R(t)=-0.037 t^{4}+1.414 t^{3}-19.777 t^{2}+118.696 t-205.332

where R represents the revenue in millions of dollars and t represents the year, with t=6 corresponding to 2006. Over which intervals is the revenue for the company increasing? Over which intervals is the revenue for the company decreasing? These questions, along with many others, can be answered by examining the graph of the polynomial function. We have already explored the local behavior of quadratics, a special case of polynomials. In this section we will explore the local behavior of polynomials in general.

Source: Rice University, https://openstax.org/books/college-algebra/pages/5-3-graphs-of-polynomial-functions
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