Solving Polynomial Equations Day 2

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Objective

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

10 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 Solving Polynomial Video Narrative explains this lesson’s Warm up-solving polynomial equations day 2 which asks students to share any strategies they used to solve polynomial equations.    

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

Finding Possible Rational Roots of Polynomial Equation

15 minutes

The goal for this portion of the lesson is for the students to recognize the pattern happening when you multiply a bunch of binomials to form a polynomial.  In the previous lesson, we looked at the how the constant of a polynomial was the product of the constants in each binomial and then used that fact to find the possible rational roots for polynomials with a leading coefficient of 1.

In this lesson, we are going to add a leading coefficient.  I start this by asking the students to determine how the problem, 2x3+9x2+7x-6=0, is different from the problems in the last lesson (has a leading coefficient greater than one) (Math Practice 7).  Once that has been determined, I tell the students that this puts monkey wrench into yesterday’s method.  I put up the factorization of the polynomial and have the students write down the roots of this equation.  Next, I remind them that yesterday the product of the zeros made the constant term.  We multiply the three zeros and have a discussion on how this is different.  What we are really doing is building the traditional p/q without adding in those variables.  

Next, a finalized method needs to be produced.  In pairs, the students come up with a method and then we finalize one as a class (Math Practice 2).  This will look very similar to p/q without those specific variables.  I highlight any different strategies here but ensure that the class has created something that is meaningful to each student. 

Finally, the students practice this new skill by listing the possible zeros for a couple of polynomials.  I add more if necessary.  

Polynomial Equations Guided Practice

25 minutes

 The remainder of the lesson is a set of Guided Practice Problems.   These increase in complexity throughout the lesson.  Please watch my video on Guided Practice Strategies.  

Homework Assignment

1 minutes

This Homework begins with 3 problems that simply ask the students to list the possible roots without solving.  This is a key skill for the success of the lesson.  The remaining portion of the lesson offers polynomial equations with a leading coefficient that is greater than one.  These problems may have both rational and possibly irrational or imaginary roots, and increase in complexity.  Please feel free to alter the assignment to fit your students.  

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 students to list the possible roots of a polynomial equation with a leading coefficient greater than one.  This is a key concept of the day’s lesson.