Objective

Warm Up

Structured Discussion

Notes on Factoring Higher Order Polynomials

Independent Practice

Exit Ticket and Homework

Seeing Structure in Expressions - Factoring Higher Order Polynomials

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Objective

SWBAT extend their quadratic factoring skills in order to write polynomial expressions in factored form.

Big Idea

Like quadratic expressions, some higher order polynomial expressions can be rewritten in factored form to reveal values that make the expression equal to zero.

Lesson Author

Colleen Werner

Ipswich, MA

Grade Level

Eleventh grade

Subjects

factor theorem

Math

factoring polynomial expressions

factorable higher-order polynomial equations

Standards

HSN-CN.C.9

(+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

HSA-SSE.A.2

Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).

HSA-APR.A.1

Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

HSA-APR.B.2

Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x - a is p(a), so p(a) = 0 if and only if (x - a) is a factor of p(x).

HSA-APR.B.3

Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

HSA-APR.C.4

Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 - y2)2 + (2xy)2 can be used to generate Pythagorean triples.

HSA-APR.D.6

Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

HSF-IF.C.7c

Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

MP1

Make sense of problems and persevere in solving them.

MP2

Reason abstractly and quantitatively.

MP3

Construct viable arguments and critique the reasoning of others.

MP4

Model with mathematics.

MP5

Use appropriate tools strategically.

MP6

Attend to precision.

MP7

Look for and make use of structure.

Warm Up

15 minutes

To begin our second unit on polynomials, I ask my students to warm-up by performing some polynomial multiplication problems in Warm up Multiplying Polynomials. Students practiced this skill in the previous unit and here they work with some "special products" that will introduce them to methods of factoring the sum of two cubes, difference of two cubes, and other common factoring patterns [MP7].

Structured Discussion

15 minutes

Connections to Computer Science

Because today is the first day of our third unit, I take a few minutes to provide students with an overview of the unit. I remind them of the formal definition of a polynomial and point out that they are now very familiar with first and second degree polynomials. In this unit, they will become familiar with the behavior of polynomials with degree three or more.

After the general overview, I focus in on what they should be able to do by the end of the day's lesson: factor some special polynomials that have a degree greater than 2.

After the introduction, I ask my students to pair up with a student near them to compare answers to the Warm-up. After they have agreed on the answers I invite them to find some problems that seem to turn out in a similar way. After 5 minutes or so, I use a random name generator to select a few students to share their findings with the class. I hope that students will see that problems #1 - 4 all turn out as the sum or difference of two cubes, 5 and 6 are both the difference of two squares and 7 and 8 are perfect square trinomials.

I then ask students to perform the following multiplication problems in their notebooks:

(a-b)(a²+ab+b²)

(a+b)(a²-ab+b²)

I ask a volunteer or two to articulate what it means to factor. I have asked this question several times before and hope that students can easily relate factoring to setting up an expression as a multiplication problem. We then discuss the special form of the two multiplications problems they completed - all the middle terms cancel leaving just the sum or difference of two cubes. Through questioning, I lead the discussion to the idea that factoring a sum or difference of two cubes is the reverse process of the multiplication they did above.

I discuss the idea that in later topics studied in Algebra 2, Precalculus and Calculus, the process of factoring a polynomial will be a "sub-procedure" in other, larger problems. I invite students who have studied computer programming to explain the concept of a sub-procedure to the other students in the class. This can lead to a great discussion about the role of computational thinking in math class if there are enough programmers in the room [MP2]!

Notes on Factoring Higher Order Polynomials

15 minutes

MP7 in Polynomials

Math Practice 7 in Polynomials.mp4

I ask my students to now take some formal notes on factoring that they can refer to later.

Factoring Higher Order Polynomials

1. Special patterns

Difference of Two Cubes: a³-b³=(a-b)(a²+ab+b²)

Sum of Two Cubes: a³+b³=(a+b)(a²-ab+b²)

Difference of Two Squares: a²-b²=(a-b)(a+b)

Perfect Square Trinomial: a²-2ab+b²=(a-b)²OR a²⁺2ab+b²=(a+b)²

I follow these notes with a few examples of using these patterns to factor.

I use a similar approach with two other types of factoring: quadratic patterns and factoring by grouping. I ask students to complete the multiplication problems on the back of Warm up Multiplying Polynomials, pair up to discuss similarities in problem types and then share their findings with the group. In their notebooks, they then add notes on the second two types of factoring.

2. Quadratic patterns

When possible, make a substitution so that a higher order polynomial can be rewritten as a quadratic trinomial. I call this a "u-substitution" to connect it to the terminology they will eventually use in Calculus.
I demonstrate this technique to students and then write out a few examples of using these patterns to factor.

3. Factoring by grouping

When a polynomial has an even number of terms, it is sometimes possible to create groups of terms that have a common factor.
Factoring out this GCF can be helpful in writing the expression in factored form.
I demonstrate this technique to my students and remind them that they already used this method as part of the process of factoring a quadratic trinomial. I write out a few examples of factoring by grouping.

In general, I hope that by the end of this lesson, my students will understand that in order factor a polynomial expression, we need to figure out the structure of the expression and then use the structure to rewrite it in a form that better illustrates factors or terms. I find that this lesson provides a good opportunity to talk with my Algebra 2 students about Mathematical Practice 7.

Independent Practice

40 minutes

After introducing today's new content, I ask students to work in table groups to complete WS Factoring High Polynomials. This worksheet is a collection of conceptual questions together with practice in factoring and solving polynomials. While I have not explicitly taught my students how to solve solve polynomial equations by factoring, I anticipate that they will connect their new factoring skills to the method they used in the previous unit of applying the Zero Product Property to determine solutions [MP1]. If they are not able to make this connection, I will help them individually at their tables, using the 3 Cup System to determine which groups need instruction.

When groups have had ample time to complete the worksheet, we go over the answers as a class and take time to discuss any problems that were difficult.

Exit Ticket and Homework

10 minutes

To make sure that students have reached the day's goal of being able to rewrite special higher order polynomials in factored form, I send them Factor Solve Polys Quick Poll, which includes questions about factoring and solving polynomials. I discuss how I use quick polls for formative assessment in the video Formative Assessment - Quick Poll. I show the results of this poll to the class so that we all have information about how the class is progressing in their understanding of factoring higher order polynomials [MP7].

For homework, students will complete a factoring puzzle, Factoring Droodle. This includes practice in all the forms we studied today along with a riddle to solve.