##
* *Reflection: Connection to Prior Knowledge
Factoring Cubic Equations - Section 3: Making Use of Structure

As we begin factoring cubic equations, it's important to build on the students prior knowledge. From Algebra 1, I find that many of my students are familiar with (and like to use) a "box" or "array" method.

At several points during this class, I'll take some time to explain in detail how this method can be applied to cubic functions. The most important point is that the array must now change from a 2 x 2 box to to a 2 x 3 box. This is due to the fact that a cubic must be divisible into one linear and one quadratic factor. The linear factor may have up to two terms, while the quadratic may have up to three terms.

You can see one example of this method worked out on my whiteboard here and another here.

*Connection to Prior Knowledge: Methods of Factoring*

# Factoring Cubic Equations

Lesson 4 of 13

## Objective: SWBAT factor a sum or difference of two cubes. SWBAT use the distributive law to multiply a binomial by a trinomial.

## Big Idea: The factored form is just as useful for solving and graphing cubic polynomials as it was for quadratics!

*45 minutes*

#### Getting Started

*15 min*

In The Biggest Box we created a cubic polynomial to model the volume of a box. We found that the graph of this function had three x-intercepts. We also found that these x-intercepts could be predicted based on the factored form of the equation. So, the factored form is just as useful for solving and graphing cubic polynomials as it was for quadratics (**MP 7**). I will begin today's class with a brief discussion to draw out these points.

I plan to tell students that we'll begin by practicing some expansion of cubic polynomials before we move on to factoring. Then I write the six equations from section 1 of Factoring Cubics on the board and ask them to quickly use the distributive law to re-write them in expanded form. This should take about 10 minutes, and I encourage them to check their answers with one another along the way.

#### Resources

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#### Observing Patterns

*10 min*

Everyone should have noticed that in a number of cases the expanded form is very brief. In fact, it simplifies to a mere sum or difference of two cubes! (I intentionally include some other cases so they don't get the impression that this *always* happens.) As students finish, I'll ask one of them to put his or her solution on the board next to the factored form. By the time our 10 minutes is up, we should have solutions to almost all of them written on the board and ready for discussion.

We'll focus on the examples of sums/differences of cubes and try to determine the pattern inherent in the factors. First, we have a linear and a quadratic factor. Second, the signs of the various terms follow a definite pattern. Third, the coefficients on each term follow a definite pattern.

I will formalize these patterns on the whiteboard under the heading: "How to Factor a Sum or Difference of Two Cubes". I'll also remind the students to be sure to copy this down into their notes!

#### Resources

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#### Making Use of Structure

*20 min*

How can we *use* this pattern? For a class example, I'll use the equation x^3 - 64. Supposing our rule for factoring a difference of two cubes is valid, what would the factors be. Once the class has given me the factors, we'll multiply them carefully to check.

The remainder of the lesson will be spent with the students working individually or in small groups to determine the factored form of the equations in section 3 of the resource.

I will spend my time helping students to apply the patterns we have just discussed. This takes special care whenever the coefficient on *x*^3 is something other than 1. Very few students will recognize on their own that 8*x*^3 is equivalent to (2*x*)^3. (Unless, of course, this is familiar to them from previous experience factoring the difference of two squares. I'd be sure to bring up this connection explicitly at some point.)

I will also point out to students that once they've reduced the equation to one linear & one quadratic factor, they should attempt to factor the quadratic, too.

Finally, I intentionally included some equations that do *not* fit this simple pattern. In this case, I've chosen functions with three integer roots to keep things simpler. These ones may pose a healthy challenge to your stronger students, and you can feel free to give lots of hints to those who are struggling. (**MP 1**)

#### Resources

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- UNIT 1: Modeling with Algebra
- UNIT 2: The Complex Number System
- UNIT 3: Cubic Functions
- UNIT 4: Higher-Degree Polynomials
- UNIT 5: Quarter 1 Review & Exam
- UNIT 6: Exponents & Logarithms
- UNIT 7: Rational Functions
- UNIT 8: Radical Functions - It's a sideways Parabola!
- UNIT 9: Trigonometric Functions
- UNIT 10: End of the Year

- LESSON 1: Intro to Power Functions
- LESSON 2: Applications of Power Functions
- LESSON 3: The Biggest Box
- LESSON 4: Factoring Cubic Equations
- LESSON 5: A Gallery of Cubic Functions, Day 1 of 2
- LESSON 6: A Gallery of Cubic Functions, Day 2 of 2
- LESSON 7: The Dynamic Cubic Function
- LESSON 8: The Factor Theorem & Synthetic Substitution
- LESSON 9: Graphs of Cubic Functions
- LESSON 10: Graphs of Cubic Functions, Day 2
- LESSON 11: The Lumber Model Problem
- LESSON 12: Cubic Equations Practice
- LESSON 13: Cubic Equations Quiz