Introduction to Polynomial Equations

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Students will be able to use roots to write and solve polynomial equations.

Big Idea

How many polynomial equations can have the same answers, find out in this lesson.

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 Video Narrative explains this lesson’s Warm Up- Solving Polynomial Equations Day 1 which asks students to solve a quadratic equation.

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

Finding Polynomials Given Roots

20 minutes

This goal of this portion of the lesson is for students to be able to take roots and create a polynomial equation.  This concept will help build the idea of solving polynomial equations in the next lesson (Math Practice 7). 

The first task gives the students the roots 5 and -7.  The students begin look at this problem independently while I wander around and give hints as necessary.  They saw this type of problem in the quadratics unit so most should feel confident.  If it seems like tons of students are struggling, I stop them all for a moment and give some reminders. 

Once we have discussed it as a class, I ask the students if this is the only polynomial with these particular roots which they discuss as partners.  The goal is for students to realize that there are an infinite number of multiples that will have the same roots (Math Practice 7).

The next problem is a bit more involved.  It gives the students three roots rather than two.   I like to use an array to multiply problems like this.  Please watch my video on multiplying with an array. 

The next problem gives the students an imaginary solution (Math Practice 1).  I give this problem to the students to try.  Once they have looked at it for a minute, I let them know that something is missing.  They talk about it in pairs and then we discuss it as a class.  We review conjugates (which they saw in the Radical Functions unit) focusing on the fact that conjugates are a difference of squares and are useful since the root or imaginary number cancels out when they are multiplied. 

We finish with two additional examples using roots.  The final example gives the root 1 + √2 which  may give the students trouble.  I find that often students will forget to turn the conjugates into a polynomial.  I walk around and correct any misunderstandings.  It is problems like this where the array multiplication is extremely useful.  


Introduction to Solving Polynomial Equations

20 minutes

Now we move on to an introduction to solving polynomial equations.  The first problem is a quadratic equation.  They just did problems like this last unit as well as the warm up.   They can solve it whichever way they feel most confident. 

The next problem is x3 + x + 10 = 0.  I give them a chance to try this problem.  Some will plow right in to the quadratic formula.  I stop them after a minute or two and ask them if they have any concerns about this problem.  At least one student will mention that this isn’t a quadratic (Math Practice 7).  I then bring up the statement on the PowerPoint that gives one of the three solutions and allow them time to talk to their partner about what they should do with this information.  Since this is so new, I may have to model this one.  I do like to give students a head start.  I don’t just plow through the problem myself; rather I give them an opportunity to figure out what to do at each step.  The goal is that they realize that once they divide the binomial x+2 out of the trinomial they are left with a quadratic that should be easy to solve. 

After one problem practicing this skill, we move up to a fourth degree problem that tells them that -2 is a root twice.  They have the opportunity to struggle with this one a bit (Math Practice 1).   Some should figure out that you can divide it out twice and then end up with a quadratic.  It may take some scaffolding.  A good question I may ask to aid this is “Now that you divided x+2 out of the polynomial, you have a cubic left.  We know that -2 is a root twice, what can we do to get down to a quadratic?” 

Homework Assignment

1 minutes

This Homework offers four practice problems for writing equations given  a polynomial and four problems that ask student to find the roots of a cubic given one of the roots.  There is also a fourth degree problem that provides a root with a multiplicity of two and asks the students to find the remaining two roots.  The final problem is an extension.  It asks the students to determine how many times a give root occurs in a fifth degree polynomial.

This assignment was created with Kuta Software, an amazing resource for secondary 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.  

This Exit Ticket asks students to write two polynomial equations with roots -2, 3, and 3. This will evaluate whether they understand both how to find polynomial equations using roots and that there is an infinite many with the same roots.