Solving Polynomial Equations Day 1

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Students will be able to solve polynomial equations with both rational and irrational roots that have at a leading coefficient of one.

Big Idea

Students build a model for finding the solutions to polynomial equations, no magic (or incomprehensible) p/q method here.

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- Solving Polynomial Equations Day 1 which asks students to analyze the work of a fictional students to decide if it is accurate.

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

Finding Possible Rational Roots of Polynomial Equation

20 minutes

During the last lesson, we discussed how the factors of numbers and polynomials represented the same thing.  We are going to review it again today.  The students find the prime factorization of 714.  Once they have completed this, I show them a polynomial and its factorization, poining out that each parenthesis represents a number. 

The next problem is an un-factored polynomial to solve.  In the last lesson, the students learned that if they divide a known root out of a cubic, they would end up with a quadratic that they could solve.  I give the students an opportunity to try this problem with no preparation other than referring back to the previous day.  Notice that I didn’t give them the p/q method of finding roots.  This is a great example of Math Practice 1.  Some students will eventually (or quickly depending on their luck) find a root.  Is students seem really stuck, I might ask the class if anyone found a root that.  The goal is that most or all of the students found a way to get the solution. 

Once we go over the problem as a class, I ask the students what they think about this method (Math Practice 3).  This is an important connection to why algebraic thinking and patterning is important.  I let them know that there is a better method.  Please watch my Video Narrative on the p/q method to see my reasoning behind the pacing in this lesson.  I note to the students that the constant terms in each binomial multiply to the constant term in the extended polynomial. 

In the next problem, I ask the students to identify what possible numbers will multiply to 40 (the constant term of the polynomial).  It is also important to note the number of zeros that this problem has. It will be obvious to some that this is awesome because you don’t have to try anything that isn’t a factor of 40.  Some may miss this point so I make sure it comes up.  I ask the students if anyone came up with a good method to not miss any of the possibilities.  

Polynomial Equations Guided Practice

20 minutes

The remainder of the lesson is a set of Guided Practice problems.   I have included a 4th degree polynomial as well as some with two irrational solutions.  

Homework Assignment

1 minutes

This Homework has a variety of polynomial equations with both rational and irrational roots.  The final question is a critical thinking problem aimed at assessing their knowledge of the structure of a polynomial equation (Math Practice 7).  

Exit Ticket

2 minutes

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

This Exit Ticket asks students to list the possible roots of a polynomial equation with a leading coefficient of one.  This is an important piece of knowledge that will help the students in the next day’s lesson.