Polynomial Long Division and Solving Polynomial Equations
Lesson 3 of 15
Objective: SWBAT relate polynomial long division to long division with integers and understand the Remainder Theorem.
I begin this lesson by asking students to take out their homework, which was a collection of long division and prime factorization problems. I grade this work with the homework rubric while they work on the warm up, Warm-up Factor GCF then Quadratic. This is a factoring worksheet that I made so that my students could get used to the process of factoring out a GCF and then continuing with another method.
For the first 4 problems in today's Warm Up, factoring out the GCF results in a quadratic expression. I remind students that factoring the quadratic expression can be thought of as a sub-procedure. I connect this to earlier conversations we have had about sub-procedures in computer science [MP2]. The final two expressions produce a cubic expression when the GCF is factored out. I included these to work in some review of factoring by grouping.
I begin this discussion by writing this question on the board:
"How do we solve polynomial equations? "
I ask students to think for a few moments, turn to a partner and discuss, and then share ideas with the class. I invite them to use the board if they want to show an example.
From this discussion, students will generally come up with examples of solving quadratic equations. Hopefully the warm-up will have also encouraged some of them to show examples of factorable higher-order polynomial equations. I give hints as necessary so that the discussion includes cubic and linear equations. I verbally summarize the techniques that students have shared. I do not expect students to take notes yet, although some may choose to.
I now ask students to consider two additional equations.
"How do we solve these polynomial equations? "
I expect that my students will reason that the first equation can be solved with the Zero Product Property (ZPP). They will also likely know that they should try to factor the second equation so that they can apply the ZPP, however they will most likely not be able to find a way to factor this expression.
When they are stuck on this, I will offer a hint: one of the factors of the expression is x-2. I give my students a few minutes to figure out they might use this hint [MP7].
Notes on Long Division
I provide my students with very detailed notes on how to divide one polynomial by another using long division. Because my students generally prefer to watch me write out the steps rather than using a Powerpoint presentation, I typically write the examples on the whiteboard and then post Polynomial Long Division on Edmodo for anyone that wants to work through the examples a second time.
After teaching my students the process of long division, I return to the example from the previous section: (6x2-25x2+21x+10) ÷ (x-2)
I show students that the process of division enables me to rewrite
6x2-25x2+21x+10 as (x-2)(6x2-13x-5)
If we continue factoring the quadratic, we see that
6x2-25x2+21x+10 = (x-2)(2x-5)(3x+1)
which allows us to see that the solutions to the original polynomial equation, 6x2-25x2+21x+10=0, are x=2, x=5/2 and x=-1/3 [MP7].
I present division in the context of solving polynomials so that my students understand why they might need to use what they consider a long and difficult procedure. In general, I find that students are more willing to put effort into learning a process when they know if will have some application in a larger context.
I ask my students to work in table groups to complete Polynomial Division, a collection of polynomial division problems. This worksheet extends student thinking about division by asking follow-up questions about whether one polynomial is a factor of the other. This will prepare my students for the following day's lesson on the Remainder Theorem.
As students work, I provide table-level help using the 3-Cup System to identify students who need support [MP1].
To assess student progress with long division, I present students with Exit Ticket Polynomial Long Division which is a polynomial long division problem. I collect these tickets to determine how much review will be necessary before presenting the Remainder Theorem.
The homework for the evening is to complete WS Polynomial Operations Including Division. In addition to polynomial long division problems, this worksheet includes some review of polynomial classification and polynomial operations learned in the previous unit. I link the full solutions to the questions on this worksheet on Edmodo so that students can check their answers before they come to class.