Contents. How You May Use This Resource Guide

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Guide

This guide is divided into chapters that match the chapters in the third editions of Technical Mathematics and Technical Mathematics with Calculus by John C. Peterson. The guide was originally developed for the second editions of these books by Robert Kimball, Lisa Morgan Hodge, and James A. Martin all of Wake Technical Community College, Raleigh, North Carolina. It has been modified for the third editions by the author.

Each chapter in this Resource Guide contains the objectives for that chapter, some teaching hints, guidelines based on NCTM and AMATYC standards, and activities. The teaching hints are often linked to activities in the Resource Guide, but also include com-ments concerning the appropriate use of technology and options regarding pedagogical strategies that may be implemented.

The guidelines provide comments from the Crossroads of the American Mathematical Association of Two-Year Colleges (AMATYC), and the Standards of the National Council of Teachers of Mathematics, as well as other important sources. These guidelines concern both content and pedagogy and are meant to help you consider how you will present the material to your students. The instructor must consider a multitude of factors in devising classroom strategies for a particular group of students. We all know that students learn better when they are actively involved in the learning process and know where what they are learning is used. We all say that less lecture is better than more lecture, but each one of us must decide on what works best for us as well as our students.

The activities provided in the resource guide are intended to supplement the excellent problems found in the text. Some activities can be quickly used in class and some may be assigned over an extended period to groups of students. Many of the activities built around spreadsheets can be done just as well with programmable graphing calculators; but we think that students should learn to use the spreadsheet as a mathematical tool. There are obstacles to be overcome if we are to embrace this useful technology for use in our courses, but it is worth the effort to provide meaningful experiences with spreadsheets to people who probably will have to use them on the job.

Whether or not you use any of the activities, we hope that this guide provides you with some thought-provoking discussion that will lead to better teaching and quality learning.

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Chapter 7

Factoring and Algebraic

Fractions

Objectives

After completing this chapter, the student will be able to: • Multiply polynomials;

• Factor polynomials using algebraic and graphical techniques; • Simplify, multiply, divide, add, and subtract algebraic fractions; • Simplify complex fractions;

• Use graphical techniques as a tool for checking;

• Use rational and polynomial functions in problem solving.

Teaching Hints

1. ntroduce this chapter through the use of an application so that students can see the need for polynomial and rational expressions. Activity 7.1 is a good introduction for polynomial functions. It will help students to see that polynomial functions do occur in everyday life. Application Example 7.71, page 350, can be used to introduce rational expressions.

2. Have students use graphics to check multiplication of polynomials. Graph both the original expression and the new expression. If the graphs for both of the expressions are the same, then the expressions are equivalent. Using graphs to check will help students understand that for any x, multiplying two expressions only changes its looks and not its value.

3. When factoring trinomials, show students both the trial and error method and the grouping method.

Example 7.1

Use factoring by grouping to factor3x2₋₁₃_x₋₁₀_.

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Solution

Here the quadraticax2₊_bx₊_c_{is first factored as}_ax2₊_b 1x+ b2x+cwhere the productac=b1+b2.

3x2₋₁₃_x₋₁₀ _ac_{= 3}_{× −}_{10 =}₋_{30 =}₋₁₅_×_{2 =}_b1_×_b2

3x2−15x+ 2x−10 Hereb1=−15andb2= 2 3x(x−5) + 2(x−5)

(3x+ 2)(x−5)

Many students have trouble with the trial and error method and have never been ex-posed to any other method for factoring trinomials. Factoring trinomials by grouping is a method that gives students a more exact procedure for factoring. Many students who are unsuccessful with the trial and error method will be successful with the grouping method.

4. In Sections 7.2 and 7.3, have students use graphical techniques to check polynomi-als that have been factored algebraically. Make sure that students understand that graphics will only let them know if they have an equivalent expression, it will not let them know if they have factored the polynomial completely. Using graphics to check will help students see that factoring a polynomial only changes how the expression looks; it does not change the value of the polynomial for anyx.

5. Show students how to use graphics as a tool for factoring. Many students have trouble with algebraic methods of factoring. Graphical methods will give students a new look at a concept that they have probably seen many times in the past. By using a graph to factor, students will be able to see how the factors of a polynomial relate to the zeros or x-intercepts of the polynomial. (see Activity 7.2)

6. Make sure that students clearly understand the difference between factors and terms. Once students are introduced to “canceling,” they want to get rid of any term that stands in their way. Emphasize that only factors, and not terms, can be reduced. Any time you reduce factors in an example make sure that you clearly state, and show, that they are factors of the numerator and denominator. It will also help if you stick with the term reduce and not refer to “canceling” in your procedure.

7. Have students use graphics to check their results in problems involving algebraic fractions. Make sure they understand that using graphics to check will only let them know if they have an equivalent expression. It will not let them know if the expres-sion is simplified completely.

Guidelines

One of the content standards in Crossroads says:

Students will use appropriate technology to enhance their mathematical think-ing and understandthink-ing and to solve mathematical problems and judge the rea-sonableness of their results.

Students need to learn how the technology of today can be used as a tool either in con-junction with algebraic processes or to replace algebraic processes. They need to see that factoring polynomials is in itself an algebraic tool and that graphics can be used as a method of factoring or as a checking device. Students need to gain confidence in their mathemati-cal abilities. With a graphics mathemati-calculator or software they have a confidence builder at their

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Instructional Resource Guide, Chapter 7

Peterson, Technical Mathematics, 3rd edition 3

fingertips. Students like to be able to check results, and graphics will allow them to quickly check algebraic results, which will, in turn, lead to building confidence.

Although manipulation of algebraic fractions is to be de-emphasized, students should still be able to find the value of an expression containing them and to understand how functions behave around its zero.

Guidelines for Content

Increased Attention

Decreased Attention

Connection of functional behavior (such as where a function increases, decreases, achieves a maximum and/or minimum, or changes concavity) to the situation mod-eled by the function

Emphasis on the manipulation of compli-cated radical expressions, factoring, ra-tional expressions, logarithms, and expo-nents

Use of statistical software and graphing calculators

Paper-and-pencil calculations and four-function calculators

Activities

1. The Box Problem

In groups, students will create a box and from the box create a polynomial function for its volume.

2. Multiplying and Factoring Polynomials with Models

Rectangular models will be used to aid students in understanding the relationship between a polynomial and its factors.

3. Factoring with Graphs

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Student Worksheet 7.1

The Box Problem

Materials: Two sheets of 70-lb8.500 ×1100 paper per group, one sheet of posterboard2200×2800, measuring

device, scissors, and tape.

1. You are a group of engineers with a company that manufactures storage containers. Your boss has asked your group to create an open-top box out of material that measures8.500×1100.

Using a sheet of paper that is8.500×1100, create an open-top box like the one in Figure 7.1.1. Once the box is created, write a polynomial function that expresses the volume of the box in terms of the side on the cut-out square. Use a graphing

calculator or graphics software to graph the polynomial. FIGURE 7.1.1

(a) Are there any restrictions on the size of the square that can be cut out? If so, how does this relate to the graph of the function?

(b) Are there any restrictions on the volume of the box? If so, how does this relate to the graph of the function? (c) Use the graph to find what size cut-out will give the maximum volume of the box. What is the maximum volume

of the box?

2. Your group did such an excellent job of creating an open-top box that your boss now wants you to create a “briefcase” box, with lid, out of the same size material:8.500×1100. The lid should be attached to the bottom but the top and the bottom do not have to be the same size, much like Figure 7.1.2.

(a) Are there any restrictions on the size of the square that can be cut out? If so, how does this relate to the graph of the function? (b) Are there any restrictions on the volume of the box? If so, how

does this relate to the graph of the function?

(c) Use the graph to find what size cut-out will give the maximum volume of the box. What is the maximum volume of the box?

FIGURE 7.1.2

3. Your boss is very happy with the results of the briefcase box. He has just been contacted by the Late Nite Pizza Company. They need a pizza box with reinforcements on the sides. Again, your group has been chosen for the job. Your goal is to create a pizza box with reinforced sides out of a piece of cardboard (posterboard) that has dimensions

2200×2800. When you have created your box, write a polynomial function that expresses the volume of the box. (a) Are there any restrictions on the size of the square that can be cut out? If so, how does this relate to the graph of

the function?

(b) Are there any restrictions on the volume of the box? If so, how does this relate to the graph of the function? (c) Use the graph to find what size cut-out will give the maximum volume of the box. What is the maximum volume

of the box?

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Student Worksheet 7.2

Multiplying and Factoring Polynomials with Models

Rectangular models can be used to factor polynomials. Examine the figure below. You should be able to see how the polynomial was broken down into its rectangular components.

x 2 2x 2x2 _4x −3 ₋3x ₋6 2x2₊_x₋₆ ⇑ ⇑ ⇑ ⇑

Now let’s use this to factor a polynomial. First you must create rectangular components from the first and last terms of the trinomial. Then you will need to create the rectangular components needed to produce the middle term. From these you can produce the factored form of the polynomial. This process is shown below for the polynomialx2+ 11x+ 28.

x2_{+ 11x}_{+ 28} ⇓ x2 _x2 _7x 28 4x 28 x x x2 ₇ 4 28 x 7 x x2 _7x 4 4x 28 ⇓ (x+ 7)(x+ 4) ⇓ ⇓ ⇓

Exercises

Use the given rectangular models to factor the following polynomials.

1. x2_{+ 3}_x_{+ 2} x2 ₂_x x 2 + + 2. 4x2_{+ 13}_x₋₁₂ 4x2 ₋₃_x 16x −12 + + 3. 3x2_{+ 17}_x_{+ 10} + +

For the following polynomials, (a) create a rectangular model and (b) write the polynomial in factored form.

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Student Worksheet 7.3

Factoring with Graphs

The graph of a polynomial provides a lot of information about the polynomial function. It can be very useful when attempting to factor a polynomial. If the zeros (x-intercepts) of the graph can be found then a factor associated with each zero can be created. Examine the graphs of the functions shown below. Compare the values of thex-intercepts to the factors of the polynomial. Can you determine the relationship?

[−5,5]×[−10,10] f(x) =x2−x−6 f(x) = (x−3)(x+ 2) FIGURE 7.3.1 [−5,5,1]×[−20,40,5] f(x) =x3₋_x2₋₁₆_x_{+ 16} f(x) = (x−4)(x+ 4)(x−1) FIGURE 7.3.2 [−5,5,1]×[−25,20,5] f(x) =x3₋_x2₋₁₂_x f(x) =x(x+ 3)(x−4) FIGURE 7.3.3

The relationship that you are seeing is that ifpis anx-intercept of the function, then (x−p)is a factor of the polynomial.

The graph of a polynomial can also be used to find a constant that may need to be factored out in order for the polynomial to be factored completely. Examine the graphs of the previous polynomials. What do you notice about the constant of the polynomial and they-intercept on the graph? Since the constant for the polynomial and they-intercept are the same, they-intercept can be used to find what constant needs to be factored out of the polynomial. To do this first find the binomial factors from thex-intercepts of the polynomial.

Now multiply the constants in the factors together. For example, if the factors are(x+ 3)(x−4)(x−2), multiply

(3)(−4)(−2) = 24. This product should be equal to they-intercept. If it is not the same, find a factor to multiply this number by to make it equal to they-intercept. This factor is the constant for your polynomial factors. For example, if the y-intercept for the polynomial with factors(x+ 3)(x−4)(x−2)is 48 then we need to find a number to multiply 24 by to get 48. Since48 = 2(24), the constant factor for this polynomial is 2 and a complete factorization of the polynomial would be2(x+ 3)(x−4)(x−2).

Now you try it. Use the graphs of the polynomial functions given below to write a complete factorization of the polynomial. [−5,5]×[−10,10] f(x) = FIGURE 7.3.4 [−5,6]×[−10,10] f(x) = FIGURE 7.3.5

Sometimes a factor will be used more than one time. For example,x2₋₄_x_{+ 4}_{will factor to}₍_x₋₂₎2 _and_x3₋

9x2_{+ 27}_x₋₂₇_{will factor to}₍_x₋₃₎3_{. Look at the graphs of these polynomials on the next page. From a graph, what} will tell you that a factor should be squared? cubed?

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[−3,6]×[−3,8] f(x) =x2₋₄_x_{+ 4} f(x) = (x−2)2 FIGURE 7.3.6 [−1,7,1]×[−40,40,5] f(x) =x3₋₉_x2_{+ 27}_x₋₂₇ f(x) = (x−3)3 FIGURE 7.3.7

Sometimes thex-intercepts of the polynomial may not be integers. In these cases numeric approximation or a root finder on a graphing utility may need to be used to find thex-intercepts of the polynomial.

Now it is time for you to try some on your own.

Exercises

(a) Use a graphing utility to graph the functions below. Choose appropriate window settings to view each function. (b) Find thex-intercepts, (c)y-intercepts, and (d) write the complete factorization of each function. Check your answers for each function by graphing your factored version on the same axes with the original function.

1. y=x2₋₉ 2. y=−3x2−3x+ 6 3. y= 4x2−4x+ 1 4. y= 8x2−6x−5 5. y=x3_{+ 6}_x2_{+ 3}_x₋₁₀ 6. y=x3_{+ 5}_x2₋₁₈_x₋₇₂ 7. y=−2x3_{+ 6}_x2_{+ 44}_x₋₄₈ 8. y=x4_{+ 5}_x3₋₃_x2₋₁₃_x_{+ 10} 9. y=x3₋₆_x2₋₁₅_x_{+ 100} 10. y=−3x4−3x3+ 39x2+ 75x+ 36 11. y=x4−8x3+ 5x2+ 50x 12. y= 2x3−14x2+ 30x−18 13. y=x4₋₅_x3_{+ 6}_x2_{+ 4}_x₋₈ 14. y=x5₊_x4₋₅_x3₋_x2_{+ 8}_x₋₄ 15. y= 20x4_{+ 119}_x3_{+ 162}_x2₋₈₁_x₋₁₀₈ 16. y= 8x6_{+ 52}_x5_{+ 38}_x4₋₁₄₅_x3_{+ 88}_x2₋₁₆_x

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Answers

Student Worksheet 7.1

1. The polynomial should beV(x) =x(11−2x)(8.5−2x)

wherexis the length of a side of the square that we be cut out of each corner.

(a) Yes,0 < x < 8.5

2 = 4.25inches. Limiting the graph to these values ofxkeeps the graph above thex-axis. (b) Yes, V > 0 in.3_{. Limiting the graph to these values}

ofV keeps the graph above thex-axis. The answers to both (a) and (b) keep the graph in the first quadrant. (c) x≈1.58in. andV ≈66.15in.3

2. The polynomial should beV(x) = 2x(11−4x)(8.5−

2x)wherexis the length of a side of the square that we be cut out of each corner.

(a) Yes,0< x < 11₄ = 2.75inches. Limiting the graph to these values ofxkeeps the graph above thex-axis. (b) Yes, V > 0 in.3. Limiting the graph to these values

ofV keeps the graph above thex-axis. The answers to both (a) and (b) keep the graph in the first quadrant.

(c) x≈1.09in. andV ≈91.48in.3

3. Answers may vary depending on how you decide to re-inforce the sides of the box. One possible answer produces V(x) = x(22−4x)(28−5x)wherexis the length of a side of the square that we be cut out of each corner. (a) Yes. For the above function,V(x) =x(22−4x)(28−

5x),0 < x < 22₄ = 5.5inches. Limiting the graph to these values ofxkeeps the graph above thex-axis. (b) Yes,V > 0 in.3_{. Limiting the graph to these values}

ofV keeps the graph above thex-axis. The answers to both (a) and (b) keep the graph in the first quadrant. (c) For the above function,V(x) =x(22−4x)(28−5x),

x≈1.85in. andV ≈506.44in.3

(d) For the above function,V(x) =x(22−4x)(28−5x), the largest-diameter pizza that will be able to fit in your box is about 14.6 in.

Student Worksheet 7.2

1. x2_{+ 3}_x_{+ 2} x 2 x 4x2 ₂_x 1 x 2 x + 2 x + 1 2. x2_{+ 3}_x_{+ 2} 4x −3 x 4x2 ₋₃_x 4 16x −12 4x + −3 x + 4 9

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3. 3x2_{+ 17}_x_{+ 10} 3x 2 x 3x2 ₂_x 5 15x 10 3x + 2 x + 5 4. (a) x2_{+ 5}_x_{+ 6} x 2 x x2 ₂_x 3 3x 6 x + 2 x + 3 (b) x2+ 5x+ 6 = (x+ 2)(x+ 3) 5. (a) 2x2₊_x₋₃ 2x 3 x 2x2 ₃_x −1 −x −3 2x + 3 x + −1 (b) 2x2₊_x₋_{3 = (2}_x_{+ 3)(}_x₋₁₎ 6. (a) 6x2_{+ 7}_x_{+ 2} 2x 1 3x 6x2 ₃_x 2 4x 2 2x + 1 3x + 2 (b) 6x2_{+ 7}_x_{+ 2 = (2}_x_{+ 1)(3}_x_{+ 2)} 7. (a) 15x2₋₄_x₋₃ 3x 1 5x 15x2 ₃₅_x −3 −12x −3 3x + 1 5x + −3 (b) 15x2₋₄_x₋_{3 = (3}_x_{+ 1)(5}_x₋₃₎

Student Worksheet 7.3

1. (a) [−5,5]×[−10,10] y =x2−9 (b) x-intercepts:−3and3 (c) y-intercept:−9 (d) y= (x−3)(x+ 3) 2. (a) [−5,5]×[−10,10] y =−3x2₋₃_x_{+ 6} (b) x-intercepts:−2and1 (c) y-intercept:9 (d) y=−3(x+ 2)(x−1) 3. (a) [−5,5]×[−10,10] y= 4x2₋₄_x_{+ 1} (b) x-intercept:0.5 (c) y-intercept:1 (d) y= (2x−1)2

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Instructional Resource Guide, Answers

4. (a) [−1,2]×[−8,10] y = 8x2₋₆_x₋₅ (b) x-intercepts:−0.5,1.25 (c) y-intercept:−5 (d) y= (x−1)(x+ 2)(x+ 5) 5. (a) [−7,3]×[−12,12,2] y =x3_{+ 6}_x2_{+ 3}_x₋₁₀ (b) x-intercepts:−5,−2,1 (c) y-intercept:−10 (d) y= (x−1)(x+ 2)(x+ 5) 6. (a) [−7,5]×[−100,20,10] y =x3_{+ 5}_x2₋₁₈_x₋₇₂ (b) x-intercepts:−6,−3,4 (c) y-intercept:−72 (d) y= (x−4)(x+ 3)(x+ 6) 7. (a) [−5,7]×[−100,100,20] y =−2x3_{+ 6}_x2_{+ 44}_x₋₄₈ (b) x-intercepts:−4,1,6 (c) y-intercept:−48 (d) y=−2(x−6)(x−1)(x+ 4) 8. (a) [−6,3]×[−50,100,20] y=x4_{+ 5}_x3₋₃_x2₋₁₃_x_{+ 10} (b) x-intercepts:−5,−2,1 (c) y-intercept:10 (d) y= (x−1)2(x+ 2)(x+ 5) 9. (a) [−5,8]×[−50,120,20] y=x3−6x2−15x+ 100 (b) x-intercepts:−5,4 (c) y-intercept:100 (d) y= (x−5)2₍_x_{+ 4)} 10. (a) [−4,5]×[−50,310,30] y=−3x4₋₃_x3_{+ 39}_x2_{+ 75}_x_{+ 36} (b) x-intercepts:−3,−1,4 (c) y-intercept:39 (d) y=−3(x−4)(x+ 1)2(x+ 3)

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11. (a) [−3,6]×[−40,100,10] y=x4₋₈_x3_{+ 5}_x2_{+ 50}_x (b) x-intercepts:−2,0,5 (c) y-intercept:0 (d) y=x(x−5)2₍_x_{+ 2)} 12. (a) [−1,5]×[−40,30,5] y= 2x3₋₁₄_x2_{+ 30}_x₋₁₈ (b) x-intercepts:1,3 (c) y-intercept:−18 (d) y= 2(x−3)2₍_x₋₁₎ 13. (a) [−2,4]×[−10,30,5] y=x4₋₅_x3_{+ 6}_x2_{+ 4}_x₋₈ (b) x-intercepts:−1,2 (c) y-intercept:−8 (d) y= (x−2)3(x+ 1) 14. (a) [−3,2]×[−10,15,5] y=x5₊_x4₋₅_x3₋_x2_{+ 8}_x₋₄ (b) x-intercepts:−2,1 (c) y-intercept:−4 (d) y= (x−1)3₍_x_{+ 2)}2 15. (a) [−4,1]×[−120,150,25] y= 20x4_{+ 119}_x3_{+ 162}_x2₋₈₁_x₋ 108 (b) x-intercepts:−3,−0.75,0.80 (c) y-intercept:−108 (d) y= (x+ 3)2(4x+ 3)(5x−4) 16. (a) [−5,2]×[−100,1300,100] y= 8x6_{+ 52}_x5_{+ 38}_x4₋₁₄₅_x3₊ 88x2₋₁₆_x (b) x-intercepts:−4,0,0.50 (c) y-intercept:0 (d) y=x(x+ 4)2₍₂_x₋₁₎3