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1 Contents How You May Use This Resource Guide ii 7 Factoring and Algebraic Fractions 1 Worksheet 7.1: The Box Problem Worksheet 7.2: Multiplying and Factoring Polynomials with Models Worksheet 7.3: Factoring with Graphs Answers 9 i

3 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 factor 3x 2 13x 10. 1

4 Instructional Resource Guide, Chapter 7 Peterson, Technical Mathematics, 3rd edition 2 Solution Here the quadratic ax 2 + bx + c is first factored as ax 2 + b 1 x + b 2 x + c where the product ac = b 1 + b 2. 3x 2 13x 10 ac = 3 10 = 30 = 15 2 = b 1 b 2 3x 2 15x + 2x 10 Here b 1 = 15 and b 2 = 2 3xx 5) + 2x 5) 3x + 2)x 5) Many students have trouble with the trial and error method and have never been exposed 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 polynomials 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 any x. 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 expression is simplified completely. Guidelines One of the content standards in Crossroads says: Students will use appropriate technology to enhance their mathematical thinking and understanding and to solve mathematical problems and judge the reasonableness of their results. Students need to learn how the technology of today can be used as a tool either in conjunction 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 mathematical abilities. With a graphics calculator or software they have a confidence builder at their

5 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. Increased Attention Connection of functional behavior such as where a function increases, decreases, achieves a maximum and/or minimum, or changes concavity) to the situation modeled by the function Use of statistical software and graphing calculators Guidelines for Content Decreased Attention Emphasis on the manipulation of complicated radical expressions, factoring, rational expressions, logarithms, and exponents Paper-and-pencil calculations and fourfunction 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 Students will use graphs as a tool for factoring polynomials.

6 Instructional Resource Guide, Chapter 7 Peterson, Technical Mathematics, 3rd edition 4 Student Worksheet 7.1 The Box Problem Materials: Two sheets of 70-lb paper per group, one sheet of posterboard 22 28, 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 measures Using a sheet of paper that is , create an opentop box like the one in Figure 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 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: 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 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 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 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? d) What is the largest-diameter pizza that the Late Nite Pizza Company will be able to fit in your box?

7 Instructional Resource Guide, Chapter 7 Peterson, Technical Mathematics, 3rd edition 5 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 2x 2 4x 3 3x 6 2x 2 + 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 polynomial x x x x + 28 x 2 x 2 7x 28 4x 28 x x x x 7 x x 2 7x 4 4x 28 x + 7)x + 4) Exercises Use the given rectangular models to factor the following polynomials. 1. x 2 + 3x x x x x + 10 x 2 2x 4x 2 3x x 2 ) + + ) 16x + 12 ) + ) + ) + ) For the following polynomials, a) create a rectangular model and b) write the polynomial in factored form. 4. x 2 + 5x x 2 + x x 2 + 7x + 2

8 Instructional Resource Guide, Chapter 7 Peterson, Technical Mathematics, 3rd edition x 2 4x 3

9 Instructional Resource Guide, Chapter 7 Peterson, Technical Mathematics, 3rd edition 7 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 the x-intercepts to the factors of the polynomial. Can you determine the relationship? [ 5, 5] [ 10, 10] [ 5, 5, 1] [ 20, 40, 5] [ 5, 5, 1] [ 25, 20, 5] fx) = x 2 x 6 fx) = x 3)x + 2) FIGURE fx) = x 3 x 2 16x + 16 fx) = x 4)x + 4)x 1) FIGURE fx) = x 3 x 2 12x fx) = xx + 3)x 4) FIGURE The relationship that you are seeing is that if p is an x-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 the y-intercept on the graph? Since the constant for the polynomial and the y-intercept are the same, the y-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 the x-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 the y-intercept. If it is not the same, find a factor to multiply this number by to make it equal to the y-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. Since 48 = 224), the constant factor for this polynomial is 2 and a complete factorization of the polynomial would be 2x + 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] [ 5, 6] [ 10, 10] fx) = FIGURE fx) = FIGURE Sometimes a factor will be used more than one time. For example, x 2 4x + 4 will factor to x 2) 2 and x 3 9x 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?

10 Instructional Resource Guide, Chapter 7 Peterson, Technical Mathematics, 3rd edition 8 [ 3, 6] [ 3, 8] [ 1, 7, 1] [ 40, 40, 5] fx) = x 2 4x + 4 fx) = x 2) 2 FIGURE fx) = x 3 9x x 27 fx) = x 3) 3 FIGURE Sometimes the x-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 the x-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 the x-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 = x y = 3x 2 3x y = 4x 2 4x y = 8x 2 6x 5 5. y = x 3 + 6x 2 + 3x y = x 3 + 5x 2 18x y = 2x 3 + 6x x y = x 4 + 5x 3 3x 2 13x y = x 3 6x 2 15x y = 3x 4 3x x x y = x 4 8x 3 + 5x x 12. y = 2x 3 14x x y = x 4 5x 3 + 6x 2 + 4x y = x 5 + x 4 5x 3 x 2 + 8x y = 20x x x 2 81x y = 8x x x 4 145x x 2 16x

12 Instructional Resource Guide, Answers Peterson, Technical Mathematics, 3rd edition x x x 2 x 3x 2 2x 5 15x 10 ) ) 3x + 2 x a) x 2 + 5x + 6 x 2 x x 2 2x 3 3x 6 ) ) x + 2 x + 3 b) x 2 + 5x + 6 = x + 2)x + 3) 5. a) 2x 2 + x 3 2x 3 x 2x 2 3x 1 x 3 ) ) 2x + 3 x + 1 b) 2x 2 + x 3 = 2x + 3)x 1) 6. a) 6x 2 + 7x + 2 2x 1 3x 6x 2 3x 2 4x 2 ) ) 2x + 1 3x + 2 b) 6x 2 + 7x + 2 = 2x + 1)3x + 2) 7. a) 15x 2 4x 3 3x 1 5x 15x 2 35x 3 12x 3 ) ) 3x + 1 5x + 3 b) 15x 2 4x 3 = 3x + 1)5x 3) Student Worksheet a) b) x-intercepts: 2 and 1 c) y-intercept: 9 [ 5, 5] [ 10, 10] y = x 2 9 b) x-intercepts: 3 and 3 c) y-intercept: 9 d) y = x 3)x + 3) 2. a) d) y = 3x + 2)x 1) 3. a) [ 5, 5] [ 10, 10] y = 4x 2 4x + 1 b) x-intercept: 0.5 [ 5, 5] [ 10, 10] y = 3x 2 3x + 6 c) y-intercept: 1 d) y = 2x 1) 2

13 Instructional Resource Guide, Answers Peterson, Technical Mathematics, 3rd edition a) b) x-intercepts: 4, 1, 6 c) y-intercept: 48 d) y = 2x 6)x 1)x + 4) [ 1, 2] [ 8, 10] y = 8x 2 6x 5 b) x-intercepts: 0.5, 1.25 c) y-intercept: 5 d) y = x 1)x + 2)x + 5) 5. a) 8. a) [ 6, 3] [ 50, 100, 20] y = x 4 + 5x 3 3x 2 13x + 10 b) x-intercepts: 5, 2, 1 c) y-intercept: 10 [ 7, 3] [ 12, 12, 2] y = x 3 + 6x 2 + 3x 10 b) x-intercepts: 5, 2, 1 c) y-intercept: 10 d) y = x 1)x + 2)x + 5) 6. a) d) y = x 1) 2 x + 2)x + 5) 9. a) [ 5, 8] [ 50, 120, 20] y = x 3 6x 2 15x b) x-intercepts: 5, 4 c) y-intercept: 100 [ 7, 5] [ 100, 20, 10] y = x 3 + 5x 2 18x 72 b) x-intercepts: 6, 3, 4 c) y-intercept: 72 d) y = x 4)x + 3)x + 6) 7. a) d) y = x 5) 2 x + 4) 10. a) [ 4, 5] [ 50, 310, 30] y = 3x 4 3x x x + 36 b) x-intercepts: 3, 1, 4 [ 5, 7] [ 100, 100, 20] y = 2x 3 + 6x x 48 c) y-intercept: 39 d) y = 3x 4)x + 1) 2 x + 3)

14 Instructional Resource Guide, Answers Peterson, Technical Mathematics, 3rd edition a) 14. a) [ 3, 6] [ 40, 100, 10] y = x 4 8x 3 + 5x x b) x-intercepts: 2, 0, 5 c) y-intercept: 0 d) y = xx 5) 2 x + 2) 12. a) [ 3, 2] [ 10, 15, 5] y = x 5 + x 4 5x 3 x 2 + 8x 4 b) x-intercepts: 2, 1 c) y-intercept: 4 d) y = x 1) 3 x + 2) a) [ 1, 5] [ 40, 30, 5] y = 2x 3 14x x 18 b) x-intercepts: 1, 3 c) y-intercept: 18 d) y = 2x 3) 2 x 1) 13. a) [ 4, 1] [ 120, 150, 25] y = 20x x x 2 81x 108 b) x-intercepts: 3, 0.75, 0.80 c) y-intercept: 108 d) y = x + 3) 2 4x + 3)5x 4) 16. a) [ 2, 4] [ 10, 30, 5] y = x 4 5x 3 + 6x 2 + 4x 8 b) x-intercepts: 1, 2 c) y-intercept: 8 d) y = x 2) 3 x + 1) [ 5, 2] [ 100, 1300, 100] y = 8x x x 4 145x x 2 16x b) x-intercepts: 4, 0, 0.50 c) y-intercept: 0 d) y = xx + 4) 2 2x 1) 3

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