Graphing Polynomials - Roots and the Fundamental Theorem of Algebra

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SWBAT Produce a sketch of the graph of a higher order polynomial function using what they have learned about end-behavior, extreme values and roots of polynomial functions.

Big Idea

A good sketch of a polynomial function can be produced by considering the end-behavior, roots and y-intercept of a polynomial function.

Warm-up and Homework Check

20 minutes

While I circulate around the room to check homework with the homework rubric at the start of today's lesson, students answer some TI NSpire Navigator questions about end behavior: Warm-up Polynomial End Behavior is a set of five questions that assess students' understanding of end behavior and the formal notation used to express it.  

The homework packet (see WS Review and Polynomial End Behavior) from the previous lesson was substantial, so I leave a little more time than usual for going over this assignment.  

Matching Activity

40 minutes

After the warm-up, my students work on Polynomial Graph Matching.  I arrange students in pairs for this activity, choosing partners based on ability.  For each partnership, I'd like to have one student who is strong with graphing and one who is strong in algebraic manipulation.  Both skills are required for success in this activity.

Polynomial Graph Matching is a set of 20 cards with algebraic and graphical representations of polynomial functions.  I included only algebraic functions in factored form to make it easier for my students to connect the graphs to the functions.  The first time I used this activity, I printed out the cards on card stock and cut apart the cards.  I have a set of storage drawers in my closet labeled with each unit I teach, so after I make them the first time I collect then and put them in the appropriate drawer for the next time I teach the unit.  The last page of Polynomial Graph Matching is a record sheet.  I print this out on plain paper and provide one sheet to each pair.

Up to this point, we have not yet explicitly discussed the relationship between the zeros of a polynomial and the x-intercepts of its graph so the matching activity will require students to think through and discuss this relationship [MP3].  While my students work I will use the 3 Cup System to provide support where needed without being "too helpful" [MP1]. I make written notes on which students seem to understand end behavior and which students are able to connect the linear factors with the roots of the equation.  When students have matched up the nine pairs, they fill in the record sheet and turn it in to me.  I remind them to leave the matched cards in front of them for our discussion.

By lining up all the record sheets in front of me, I can quickly see which of the graphs were hard and easy for my students.  We go through each of the matches, discussing how we know which algebraic function pairs with each graph [MP2, MP3]. 

Structured Discussion about End Behavior, Roots and Multiplicity

20 minutes

After reviewing the correct pairs for the Polynomial Graph Matching Activity, we discuss how the process of finding the x-intercepts of a polynomial relates to the Fundamental Theorem of Algebra and the Remainder Theorem.  It's important to me that my students to not think of Algebra 2 as a long list of unrelated topics, so I am careful to give them time to connect new knowledge to what we have already learned. I will structure this discussion so that students can make sense of the following concepts:

  • The Remainder Theorem says that f(a)=0 if and only if f(x) divided by x-a has a remainder of zero.  This makes sense because plugging in the x-intercept into the function gives and output of zero.  All the x-intercepts come from setting the linear factors equal to zero.
  • The Fundamental Theorem of Algebra talks about the number of complex roots.  The degree of the polynomial is sometimes bigger than the number of x-intercepts, so the complex roots must not be x-intercepts. 
  • The Fundamental Theorem of Algebra says that a polynomial of degree n has n complex roots provided repeated roots are counted separately.  The "counted separately" refers to roots where the graph touches and then turns around rather than crossing through.  

This last point leads to a discussion of multiplicity, which will be a new concept for my students.  Using examples of factored polynomial functions like 

  • f(x)=(x-2)2(x-3)

After these final polynomial graph concepts have been explored, we take notes on the most efficient way of producing a good sketch of a polynomial graph.  I advise my students to use a process something like this:

  • Think about end behavior and maybe draw in some small arrows to remind themselves
  • Calculate the y intercept and plot it on the graph
  • Factor the polynomial function completely in the real number system (if it isn't presented that way) to make it easy to determine the x-intercepts.  Plot these on the graph, noting whether the graph will touch or cross at each intercept.
  • Connect the intercepts in such a way that the graph turns out to have the expected end behavior.

Exit Ticket and Assignment

15 minutes

As an exit ticket, I ask students to produce a sketch of two polynomial functions using the strategy presented in the notes.  Exit Ticket-Graphs of Polynomials has two functions for students to graph:

  • One factorable quartic function whose factors all have a multiplicity of one
  • One factorable cubic function with one double root and a single root

I want students to reflect on the differences between even and odd functions. 

For homework, student will work on 16 questions related to the graphs of polynomial functions in WS Graphing Polynomials.  These exercises focus on my students' ability to connect the algebraic form of a polynomial function to the graph of the function [MP2].