# Lesson 7

Using Factors and Zeros

### Lesson Narrative

The purpose of this lesson is for students to use what they have learned about the relationship between the factors of a polynomial and the zeros of a polynomial. Students first use their knowledge of the structure of polynomials in factored form (MP7) to identify an appropriate horizontal range for the graphing windows for polynomials given an equation, and then write a possible equation for a polynomial with specific horizontal intercepts, turning around the work they have done previously. Students also revisit the kind of vertical scaling they encountered in the previous lesson to find an appropriate vertical range for a graphing window. The work students do in this lesson helps them learn how to apply the connections they have been making between graphs and equations.

An important takeaway of this lesson is that a polynomial with a factor of $$x-a$$ has a zero when $$x=a$$. Students will continue to use this idea throughout future lessons as they build up to the Remainder Theorem, which allows us to prove that a polynomial with a zero at $$x=a$$ must have $$x-a$$ as a linear factor.

### Learning Goals

Teacher Facing

• Explain how to find an appropriate graphing window for a function in factored form.
• Generate a possible expression for a polynomial function given the horizontal intercepts of the function.

### Student Facing

• Let’s write some polynomials.

### Required Preparation

Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has their own device. (Desmos is available under Math Tools.)

### Student Facing

• I can write an expression for a function that has specific horizontal intercepts.