Factoring Completely

By: Allison Trask

Throughout this document anytime you see ^ it means that the term is raised to the power following this symbol.

This lesson is based from the textbook Southwestern Algebra I: An Integrated Approach.
Gerver, Sgroi, Carter, Hansen, Molina, Westegaard. (1998). Southwestern algebra 1: An integrated approach. Cincinnati: Southwestern Educational Publishing.

Lesson Title: Time to Factor!
Grade Level: 7th or 8th grade
Course Title: Algebra I
Time Allotted: 1 class period
Number of Students: 24 students
Extra Information About Students: None
Day 5

Goals and Objectives:
According to the NCTM Principles and Standards of Mathematics, the following standards are met in this lesson:
1. To understand numbers, ways of representing numbers, relationships among numbers, and number systems.

Use factors, multiples, prime factorization, and relatively prime numbers to solve problems.
Develop meaning for integers and represent and compare quantities with them.

2. To understand meanings of operations and how they relate to one another.

Understand the meaning and effects of arithmetic operations with fractions, decimals, and integers.
Use the associative and commutative properties of addition and multiplication and the distributive prepoerty of multiplication over addition to simplify computations with integers, fractions, and decimals.
Understand and use the inverse relationships of addition and subtraction, multiplication and division, and squaring and finding square roots to simplify computations and solve problems.

According to the NCSCOS, the following standards are met in this lesson:
1. Competency Goal 1: The learner will perform operations with real numbers and polynomials to solve problems.

Operate with real numbers to solve a variety of problems.
Operate with polynomials.
Factor polynomials.

Materials Needed and/or Use of Space:

Calculators for each group of students

Motivational Activity:

The point of this activity is to remind students about prime factorizations and remembering what prime numbers are. The point of this activity is to get the students warmed up and ready for factoring polynomials. Give the students 5-10 minutes to complete this activity.

1. Have the students break into groups of 3 or 4. You may want to break them up into set groups prior to the activity.

2. Each group should have a calculator to use.

3. Remind the students that a prime factorization of a number shows the number as a product of ONLY prime numbers. For example, the prime factorization of 24 is 2*2*2*3 OR (2^3)*3.

4. Give the students the Prime Factorization worksheet for this activity. Students may use calculators if needed. Remember, this worksheet is given to the students as a review of prime factorization and as a preview to factoring, not as a lesson in itself.

5. Upon completion of this worksheet, have the groups of students compare their answers with another group. ask the students the following questions:

Can you describe what you did to find the factorization in problem 1?
Which method appears to be the most efficient?

Describe that today we are going to learn how to perform two or more types of factoring on the same polynomial. The reason that we are going to do this is so that we can understand how to factor polynomials completely in order to solve problems in geometry, as well as real world applications.

Lesson Procedure:

The past few days we have discussed different ways to factor polynomials. For example, how would we factor 3x^3-12x^2+6x?

Yes, the answer would be equal to 3x(x^2-4x+2). How did we get this answer?

We have also learned how to factor trinomials, as well as perfect square binomials, as the product of two binomials. We have even looked at factoring the difference of two squares as the product of two binomials.

For the following examples, be sure to give the students ample wait time to complete the problems. Depending upon the students and how much time you have to teach the lesson, do the following review problems either on the board, as individual board work, or let the students complete the problems in their notes. This part is up to the individual teacher since it is another quick review for the main part of the lesson.

In your notes, figure out the factorization of x^2+4x+3. This is equal to (x+3)(x+1).
Can you figure out how to factor 4x^2-28x+49? This is equal to (2x-7)^2.
Let's try this one together: 16x^2-25 = (4x+5)(4x-5).

We do not consider a polynomial to be "factored" until we have factored it completely. A polynomial is factored completely only when it is expressed as the product of a monomial and one or more polynomials expressions that cannot be factored any further. Sometimes we have to factor polynomials more than once before they are factored completely. Let's look at a few examples together and factor completely!

Example 1: Factor: 2x^2-14x+24.
The first thing that we need to look for is a common monomial factor. Does anyone see something that all of the terms have in common? Yes, we see that the GCF in this problem is 2. So, let's factor out a 2.

When we factor out a 2, we are left with:
2x^2-14x+24 = 2((x^2)-7x+12) Now, we can just factor the trinomial that we have.
= 2(x-3)(x-4)

We can check this answer by multiplying all of the factors. Let's do this just to prove that we have arrived at the correct answer. Are there any questions before we move on?

Again, you can do this depending upon how much time is left in the class period. There may not be enough time so you may need to move on to the next example.

Example 2: Factor: 5(x^2)y+20x^2-45y-180.
Again, the first thing that we need to look for is the GCF for all numbers. Do you see a common factor? Yes, the GCF is 5.

When we factor out a 5, we are left with:
       5(x^2)y+20x^2-45y-180 = 5((x^2)y + (4x^2) - 9y - 36)              Now we need to group our terms.
                                              = 5(((x^2)y + 4x^2) - (9y + 36))           Do we have a common factor in each group? YES!
                                              = 5(x^2(y + 4) - 9(y + 4))                      Because we have a commong factor of (y+4), we can factor this out and                                                                                                                combine what is left.
                                                = 5(y + 4)(x^2 - 9)                                 Can we do anymore factoring? YES! We have a difference of two squares!
                                                = 5(y + 4)(x + 3)(x - 3)
We cannot factor any further so we are finished! Great job!

Does anyone have a question before we move on to a word problem?

Example 3: Let's try a word problem together.
The volume of a rectangular prism is 3ab^2 - 6ab - 45a. Find the dimensions of this prisms in terms of a and b.

So the question is, what does it mean to have our dimensions in terms of a and b? Any ideas? (Give students wait time and see what answers you get. Go from there!) This means that we need to factor out factors of a and factors of b; we need to find the dimensions of our rectangular prism with our dimensions involving a and b. Now we're ready to factor!

Do we have a common factor? Yes, 3a.

3ab^2 - 6ab - 45a = 3a(b^2 - 2b - 15) Now we can factor our trinomial.
= 3a (b - 5)(b+3)

So we have now found the dimensions of our rectangular prism. The dimesions are 3a, b - 5, and b + 3.

Good job today! You guys are going to be pros at factoring completely :) I am now going to give you a worksheet to practice factoring completely. The main way to become better at factoring completely is to practice.

For homework, finish the worksheet if you do not complete it during class today. Also, complete the journal entry on the board. If you have any questions or comments while working on your homework, please raise your hand and I will come help you.

Journal Entry: We started off today with reviewing prime factorization. How is this similar to factoring completely? What are the different types of factoring? How do you know when a polynomial has been factored completely?

Extension:

If the students complete their homework worksheet before class is finished, have those students make up 5 problems of their own and then factor them completely. Tell these students that if their problems are fair and just, that you may use their problems on an upcoming quiz or test. This will give the students the opportunity to engage in mathematics by making up their own problems. This way, the students will feel that they had a say in their upcoming quiz or test. The students will also be using the problem solving skills, as well as communication skills when completing this activity.

Closure:

As you can see, we have learned how to factor completely today. We have used all of the skills that we have learned about factoring thus far and have combined them all into one main category. This is a skill that you will use in future mathematics classes, as well as in real world applications. In the next few days, we will learn how factoring is used in the real world. Good luck with your homework tonight and with practicing how to factor completely!

Assessment:

Students will complete the journal entry. Their completion of this journal will show the teacher whether or not they understood what they learned. This activity also teaches the students more about mathematical communication.
The students will also have a worksheet to complete. If the worksheet is not completed in class, it is to be finished as homework. This worksheet will count as a homework grade.
The prime factorization worksheet will count as classwork for the students.
The students group work will count as part of their conduct/behavior grade.

Evaluation of Lesson Upon Completion:

What would you do differently?
What would you do the same?
Was this a good lesson - why or why not?

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