Factored and Standard Forms of Polynomials (Lesson 2.7)
Unit 2 Day 9
CED Topic(s): 1.11
Describe the advantages of writing a polynomial in factored form versus standard form.
Convert polynomials from factored to standard forms and vice versa.
Explain why when (x-k) is a factor of a polynomial, x=k is a zero of the polynomial
Find all zeros of a polynomial function by hand or using technology.
Quick Lesson Plan
In the next three lessons, students will be introduced to a variety of strategies for manipulating polynomial and rational functions with a focus on how the different forms are equivalent but reveal various features of the function. Today’s activity focuses on the standard and factored form of a polynomial. The activity relies heavily on the area model as a way to see the distributive property and as a way to divide polynomials. If your students are unfamiliar with the area model, ask them to stop after question 1 and then as a whole class discuss where the like terms appear in the area model, before having them complete the rest of the activity. In question 3 students work backward from the area model by applying what they just noticed about the like terms in the expansion. Note that we do not pre-teach students how to divide using the area model! The process is rather intuitive and it’s fun to see all the lightbulbs turn on as students figure out on their own how to determine the other factor.
Question 5 gives students another opportunity to connect factors with solutions with x-intercepts. One of the goals of today’s lesson is for students to identify why one might want to write a polynomial in standard form and why one might want to write a polynomial in factored form. The ability to see the zeros of a function is a huge plus to factored form!
Note that questions 6 and 7 are simply about determining if the expression x=3 is a factor, not about actually writing out the quotient with the remainder. This will be the topic of tomorrow’s lesson! Use simple numerical examples to show the relationship between factors and dividing evenly.
What does each box in the area model represent? Why do we use an area model?
Is 3 a factor of 183? What does that mean? What is the other factor? How did you figure it out?
If I know the factors of a polynomial, can I find its zeros?
If I know the zeros of a polynomial, can I find its factors?
If x+3 and x^2+x+2 were the factors, what should the product be?
In the debrief, using color to identify the like terms can be very helpful. This allows students to make sense of the area model and be able to flexibly use it when expanding from factored form, or factoring from standard form, given one factor or zero.
In the Quicknotes, be sure to point out that the values of p, q, and k which represent the zeros of the function can be imaginary! They are still zeros, just not x-intercepts.
For polynomials of degree 3 or greater, we expect students to use some kind of technology to identify a real zero (unless the polynomial has a greatest common factor of x). Similarly, they could be given a table of selected values where one of the values gives an x-intercept. Once students have a real zero, they can divide by the corresponding factor, determine the other factor, and then identify the remaining zeros, whether real or complex. This process is explored in question 3 of the Check Your Understanding.