##
* *Reflection: Discourse and Questioning
Solving Polynomial Equations Day 2 - Section 2: Finding Possible Rational Roots of Polynomial Equation

In the past lesson, my students did well in developing the concept that the product of the zeros of a polynomial is equivalent to the opposite of the constant (at least where the leading coefficient is one). Today, I found that making the jump to having a leading coefficient other than one was challenging. We looked at the possible zeros but I wanted them to make connection that the product of the denominators should be equivalent to the leading coefficient. One question that seemed to make a huge difference was whether 1/5 and 3/5 could both be zeros of a polynomial with a leading coefficient of 5. I loved watching the animated conversations as students made the connection while discussing this with their partner. So often, questions like this can make the difference in conceptual understanding and can really set apart a strong Common Core lesson.

*Discourse and Questioning: Asking Deeper Conceptual Questions*

# Solving Polynomial Equations Day 2

Lesson 9 of 15

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

#### Warm Up and Homework Review

*10 min*

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.

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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, 2x^{3}+9x^{2}+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.

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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.

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#### Homework Assignment

*1 min*

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.

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#### Exit Ticket

*2 min*

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.

#### Resources

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- UNIT 1: Modeling with Expressions and Equations
- UNIT 2: Modeling with Functions
- UNIT 3: Polynomials
- UNIT 4: Complex Numbers and Quadratic Equations
- UNIT 5: Radical Functions and Equations
- UNIT 6: Polynomial Functions
- UNIT 7: Rational Functions
- UNIT 8: Exponential and Logarithmic Functions
- UNIT 9: Trigonometric Functions
- UNIT 10: Modeling Data with Statistics and Probability
- UNIT 11: Semester 1 Review
- UNIT 12: Semester 2 Review

- LESSON 1: Polynomial Long Division
- LESSON 2: Synthetic Division
- LESSON 3: Uses of Polynomial Division- The Factor and Remainder Theorems
- LESSON 4: Introduction to Polynomial Equations
- LESSON 5: Solving Polynomial Equations Graphically
- LESSON 6: Graphs of Polynomial Functions
- LESSON 7: Intercepts of Polynomial Functions
- LESSON 8: Solving Polynomial Equations Day 1
- LESSON 9: Solving Polynomial Equations Day 2
- LESSON 10: Transformation of Polynomial Functions
- LESSON 11: Modeling Volume with Polynomials Day 1
- LESSON 12: Modeling Volume with Polynomials Day 2
- LESSON 13: Polynomial Functions Review Day 1
- LESSON 14: Polynomial Functions Review Day 2
- LESSON 15: Polynomial Functions Test