Uses of Polynomial Division- The Factor and Remainder Theorems

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Objective

Students will be able to use the remainder and factor theorems to find the factors of polynomials and to solve polynomial equations.

Big Idea

Show students why polynomial division is useful in the lesson.

Warm Up and Homework Review

5 minutes

I include Warm ups with a Rubric as part of my daily routine. My goal is to allow students to work on Math Practice 3 each day. Grouping students into homogenous pairs provides an opportunity for appropriately differentiated math conversations. The Video Narrative explains this lesson’s Warm Up- Uses of Polynomial Division which asks students to write a list of steps used to do synthetic division.

I also use this time to correct and record the previous day's Homework

The Remainder Theorem

20 minutes

The goal of this lesson is to look at some of the uses of synthetic division and to prepare students to solve and graph polynomial equations and functions.  The very first task asks the students to evaluate a polynomial function at f(3).  I give the students this task with no introduction and ask them to check their answer with their partner.  We then go through it as a class.

Now I demonstrate the same problem using synthetic division without forewarning the students about what to expect.  They are pretty impressed that it turns out to be the same answer.  Some students will probably note that synthetic division is no easier than the regular method.  This is an excellent place to discuss when it would be good to use one method and when the other (Math Practice 5). 

Next, I introduce the students to the formal written Remainder Theorem.  While the key issue is that students understand the concept behind the theorem, it is good for them to look at formal written theorems.  I tell students that this is a big idea for the day and ask them to but a box or cloud around it with some hearts and flowers to show how important it is (humor).  I also encourage my students to have colored pens or highlighters to use for notes.  Please watch my video on Note Taking for more information. 

One issue here is to discuss is WHY does the Remainder Theorem work?  This will be dealt with when we get to graphing polynomial functions.

There is an additional practice problem.  I may add more as needed by my students.  

The Factor Theorem

20 minutes

Now we are moving on to the Factor Theorem.  This Theorem lays the basis for a good portion of the rest of the unit.  Often students will not connect that factors in polynomials are the same as factors of numbers. 

I have the students write a statement answering the question “What is a factor?” and then ask them to share with their partner.  We then discuss it as a class.  I try to lead the discussion to the fact that factors also apply to polynomials.  If no one brings it up, I ask the students if polynomials also have factors.  I give them a chance to talk with their partners and then ask for volunteers to share out.  Use some examples, like a simple quadratic can be helpful and link it to a number.  For example: x2 + 5x + 6 factors to (x+2)(x+3).  What if x where 1?  This would give us 12 = (3)(4).  This is also a great time to remind students why algebra is so amazing.  We have an INFINITE number of possible number combinations in one time space (Math Practice 2)

Now we discuss the other terms used for a factor and make a list as a class.  They come up with things like zero or root.   For each word that is used, I give them some perspective as to its meaning.  For example, a zero represents where a polynomial crosses the x axis.  A common place for students to get confused in mathematics is where different terminology is used for something they know how to do.  Our job is to give them exposure to as many of these terms as possible. 

Now they are given the task of determining if a binomial is a factor of a larger polynomial.  This problem will reinforce the concept of factor or zero.  I give them an opportunity to discuss and then ask for volunteers to answer the question.  Once the answer is clear, I give everyone an opportunity to finish the problem. 

Once they have an idea of how to determine if a binomial is a factor, there is an additional practice problem.  I add more as needed by my students. 

The formal Factor Theorem is now introduced.   This is the second big idea for the day.  I make sure to discuss each portion after they have copied it down.  It is important to model reading mathematical statements. 

Our final task involves a problem where i is the factor.  The students try this one with no introduction.  It is a challenging problem (Math Practice 1).  Again, I give them an opportunity to struggle with it,   walking around and giving hints.  The double distribution may aggravate some.  One strategy I use with particularly challenging problems is to walk around and announce as I see correct solutions.  Students will do a lot for praise or recognition.   We go over the problem as a class once most of the students have finished.  

Homework Assignment

1 minutes

This Homework practices both the Remainder and the Factor Theorems.  The final two problems are extensions similar to the final problem in the lesson.  These two can be added or taken away to differentiate the assignment.  Please note that this worksheet is comparable to the book assignment I am giving my students.  

This assignment was created using Kuta Software, an amazing resource for mathematics teachers.

Exit Ticket

2 minutes

I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.  

Today's Exit Ticket asks student to determine whether x – 1 is a factor of x5 – 1.