SECTION P.5 Factoring Polynomials


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1 BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page Chapter P Prerequisites: Fundamental Concepts of Algebra Technology Eercises Critical Thinking Eercises 98. The common cold is caused by a rhinovirus. The polynomial In Eercises 00 0, perform the indicated operations. describes the billions of viral particles in our bodies after days of invasion. Use a calculator to find the number of viral particles after 0 days (the time of the cold s onset), day, days, days, and 4 days.after how many days is the number of viral particles at a maimum and consequently the day we feel the sickest? By when should we feel completely better? 99. Using data from the National Institute on Drug Abuse, the polynomial 0. [(7 + 5) + 4y D C(7 + 5)  4yD 00. (  y)  ( + y) 0. C( + y) + D 0. ( + y)(  y)a + y B 04. Epress the area of the plane figure shown as a polynomial in standard form approimately describes the percentage of U.S. high school seniors in the class of who had ever used marijuana, where is the number of years after 980. Use a calculator to find the percentage of high school seniors from the class of 980 through the class of 000 who had used marijuana. Round to the nearest tenth of a percent. Describe the trend in the data. + SECTION P.5 Factoring Polynomials Objectives. Factor out the greatest common factor of a polynomial.. Factor by grouping.. Factor trinomials. 4. Factor the difference of squares. 5. Factor perfect square trinomials. 6. Factor the sum and difference of cubes. 7. Use a general strategy for factoring polynomials. 8. Factor algebraic epressions containing fractional and negative eponents. A twoyearold boy is asked, Do you have a brother? He answers, Yes. What is your brother s name? Tom. Asked if Tom has a brother, the twoyearold replies, No. The child can go in the direction from self to brother, but he cannot reverse this direction and move from brother back to self. As our intellects develop, we learn to reverse the direction of our thinking. Reversibility of thought is found throughout algebra. For eample, we can multiply polynomials and show that ( + )(  ) We can also reverse this process and epress the resulting polynomial as ( + )(  ).
2 BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page 49 Factor out the greatest common factor of a polynomial. Section P.5 Factoring Polynomials 49 Factoring is the process of writing a polynomial as the product of two or more polynomials. The factors of are + and . In this section, we will be factoring over the set of integers, meaning that the coefficients in the factors are integers. Polynomials that cannot be factored using integer coefficients are called irreducible over the integers, or prime. The goal in factoring a polynomial is to use one or more factoring techniques until each of the polynomial s factors is prime or irreducible. In this situation, the polynomial is said to be factored completely. We will now discuss basic techniques for factoring polynomials. Common Factors In any factoring problem, the first step is to look for the greatest common factor. The greatest common factor, abbreviated GCF, is an epression of the highest degree that divides each term of the polynomial. The distributive property in the reverse direction ab + ac a(b + c) can be used to factor out the greatest common factor. EXAMPLE Factoring out the Greatest Common Factor Factor: a b. ( + ) + 5( + ). Study Tip The variable part of the greatest common factor always contains the smallest power of a variable or algebraic epression that appears in all terms of the polynomial. a. We begin by determining the greatest common factor. 9 is the greatest integer that divides 8 and 7. Furthermore, is the greatest epression that divides and. Thus, the greatest common factor of the two terms in the polynomial is () + 9 () Epress each term as the product of the greatest common factor and its other factor. 9 ( + ) Factor out the greatest common factor. b. In this situation, the greatest common factor is the common binomial factor ( + ). We factor out this common factor as follows: ( + ) + 5( + ) ( + )A + 5B. Factor out the common binomial factor. Factor by grouping. Factor: a. 04 b. (  7) + (  7). Factoring by Grouping Some polynomials have only a greatest common factor of. However, by a suitable rearrangement of the terms, it still may be possible to factor. This process, called factoring by grouping, is illustrated in Eample.
3 BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page Chapter P Prerequisites: Fundamental Concepts of Algebra EXAMPLE Factoring by Grouping Factor: Group terms that have a common factor: Common factor is. Common factor is. Discovery In Eample, group the terms as follows: A + B + A4 + B. Factor out the greatest common factor from each group and complete the factoring process. Describe what happens. What can you conclude? We now factor the given polynomial as follows ( + 4 ) + ( + ) ( + 4) + ( + 4) ( + 4)( + ) Group terms with common factors. Factor out the greatest common factor from the grouped terms. The remaining two terms have + 4 as a common binomial factor. Factor ( + 4) out of both terms. Thus, ( + 4)( + ). the factorization by multiplying the right side of the equation using the FOIL method. If the factorization is correct, you will obtain the original polynomial. Factor: Factor trinomials. Factoring Trinomials To factor a trinomial of the form a + b + c, a little trial and error may be necessary. A Strategy for Factoring a b c (Assume, for the moment, that there is no greatest common factor.). Find two First terms whose product is a : (n + )(n + ) a + b + c. Î Î Î. Find two Last terms whose product is c: ( +n)( +n) a + b + c. Î Î Î. By trial and error, perform steps and until the sum of the Outside product and Inside product is b: (n +n)(n +n) a + b + c. I O (sum of O+I) Î If no such combinations eist, the polynomial is prime.
4 BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page 5 Factors of 8 8, 4, 8,  4,  Sum of Factors This is the desired sum. Factors of 8 Sum of factors 8,  78, 7 9,  79, 7 6,  6,  This is the desired sum. Section P.5 Factoring Polynomials 5 EXAMPLE Factoring Trinomials Whose Leading Coefficients Are Factor: a b a. The factors of the first term are and : To find the second term of each factor, we must find two numbers whose product is 8 and whose sum is 6. From the table in the margin, we see that 4 and are the required integers. Thus, b. We begin with ( )( ) ( + 4) ( + ) or ( + ) ( + 4) ( )( ). To find the second term of each factor, we must find two numbers whose product is 8 and whose sum is. From the table in the margin, we see that 6 and  are the required integers. Thus, ( + 6)(  ) or (  )( + 6). Factor: a b EXAMPLE 4 Factoring a Trinomial Whose Leading Coefficient Is Not Factor: Step Find two First terms whose product is (8 )( ) (4 )( ) Step Find two Last terms whose product is . The possible factorizations are () and (). Step Try various combinations of these factors. The correct factorization of is the one in which the sum of the Outside and Inside products is equal to 0. Here is a list of the possible factorizations: Possible Factorizations Sum of Outside and Inside of Products (Should Equal 0) (8 + )(  ) (8  )( + ) 85 (8  )( + ) 4  (8 + )(  ) (4 + )(  ) (4  )( + ) (4  )( + )  0 (4 + )(  ) This is the required middle term.
5 BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page 5 5 Chapter P Prerequisites: Fundamental Concepts of Algebra Thus, (4 + )(  ) or (  )(4 + ). Show that this factorization is correct by multiplying the factors using the FOIL method. You should obtain the original trinomial. 4 Factor: Factor the difference of squares. Factoring the Difference of Two Squares A method for factoring the difference of two squares is obtained by reversing the special product for the sum and difference of two terms. The Difference of Two Squares If A and B are real numbers, variables, or algebraic epressions, then A  B (A + B)(A  B). In words: The difference of the squares of two terms factors as the product of a sum and a difference of those terms. EXAMPLE 5 Factoring the Difference of Two Squares Factor: a.  4 b We must epress each term as the square of some monomial. Then we use the formula for factoring A  B. a ( + ) (  ) A B (A + B) (A B) b (9)  7 (9 + 7)(97) 5 Factor: a.  8 b Study Tip Factoring 48 as A + 9B A  9B is not a complete factorization. The second factor,  9, is itself a difference of two squares and can be factored. We have seen that a polynomial is factored completely when it is written as the product of prime polynomials. To be sure that you have factored completely, check to see whether the factors can be factored. EXAMPLE 6 Factor completely: 48 A B  9 A + 9B A  9B A Repeated Factorization 48. Epress as the difference of two squares. The factors are the sum and difference of the squared terms.
6 BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page 5 Section P.5 Factoring Polynomials 5 A + 9B A  B A + 9B( + )(  ) The factor  9 is the difference of two squares and can be factored. The factors of  9 are the sum and difference of the squared terms. 6 Factor completely: Factor perfect square trinomials. Factoring Perfect Square Trinomials Our net factoring technique is obtained by reversing the special products for squaring binomials. The trinomials that are factored using this technique are called perfect square trinomials. Factoring Perfect Square Trinomials Let A and B be real numbers, variables, or algebraic epressions.. A + AB + B (A + B) Same sign. A  AB + B (A  B) Same sign The two items in the bo show that perfect square trinomials come in two forms: one in which the middle term is positive and one in which the middle term is negative. Here s how to recognize a perfect square trinomial:. The first and last terms are squares of monomials or integers.. The middle term is twice the product of the epressions being squared in the first and last terms. EXAMPLE 7 Factoring Perfect Square Trinomials Factor: a b a ( + ) The middle term has a positive sign. A + A B + B (A + B) b. We suspect that is a perfect square trinomial because 5 (5) and 6 6. The middle term can be epressed as twice the product of 5 and (5) (56) A A B + B (A B)
7 BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page Chapter P Prerequisites: Fundamental Concepts of Algebra 6 Factor the sum and difference of cubes. 7 Factor: a b Factoring the Sum and Difference of Two Cubes We can use the following formulas to factor the sum or the difference of two cubes: Factoring the Sum and Difference of Two Cubes. Factoring the Sum of Two Cubes A + B (A + B) AA  AB + B B. Factoring the Difference of Two Cubes A  B (A  B) AA + AB + B B EXAMPLE 8 Factoring Sums and Differences of Two Cubes Factor: a. + 8 b a ( + )A  + B ( + ) A  + 4B A + B (A + B)(A AB + B ) b (4)  5 (45) [(4) + (4)(5) + 5 ] A B (A B)(A + A B + B ) (45) ( ) 8 Factor: a. + b Use a general strategy for factoring polynomials. A Strategy for Factoring Polynomials It is important to practice factoring a wide variety of polynomials so that you can quickly select the appropriate technique. The polynomial is factored completely when all its polynomial factors, ecept possibly for monomial factors, are prime. Because of the commutative property, the order of the factors does not matter.
8 BLITMCPB.QXP.0599_4874_tg //0 :8 PM Page 55 Section P.5 Factoring Polynomials 55 A Strategy for Factoring a Polynomial. If there is a common factor, factor out the GCF.. Determine the number of terms in the polynomial and try factoring as follows: a. If there are two terms, can the binomial be factored by one of the following special forms? Difference of two squares: A  B (A + B)(A  B) Sum of two cubes: A + B (A + B)(A  AB + B ) Difference of two cubes: A  B (A  B)(A + AB + B ) b. If there are three terms, is the trinomial a perfect square trinomial? If so, factor by one of the following special forms: A + AB + B (A + B) A  AB + B (A  B). If the trinomial is not a perfect square trinomial, try factoring by trial and error. c. If there are four or more terms, try factoring by grouping.. to see if any factors with more than one term in the factored polynomial can be factored further. If so, factor completely. EXAMPLE 9 Factoring a Polynomial Factor: Step If there is a common factor, factor out the GCF. Because is common to all terms, we factor it out ( ) Factor out the GCF. Step Determine the number of terms and factor accordingly. The factor has three terms and is a perfect square trinomial. We factor using A + AB + B (A + B) ( ) ( + + ) A + A B + B ( + ) A + AB + B (A + B) Step to see if factors can be factored further. In this problem, they cannot. Thus, ( + ). 9 Factor:
9 BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page Chapter P Prerequisites: Fundamental Concepts of Algebra EXAMPLE 0 Factoring a Polynomial Factor:  5a Step If there is a common factor, factor out the GCF. Other than or, there is no common factor. Step Determine the number of terms and factor accordingly. There are four terms. We try factoring by grouping. Grouping into two groups of two terms does not result in a common binomial factor. Let s try grouping as a difference of squares.  5a ( )  5a Rearrange terms and group as a perfect square trinomial minus 5a to obtain a difference of squares. ( + 4)  (5a) Factor the perfect square trinomial. ( a)( + 45a) Factor the difference of squares. The factors are the sum and difference of the epressions being squared. Step to see if factors can be factored further. In this case, they cannot, so we have factored completely. 0 Factor:  6a Factor algebraic epressions containing fractional and negative eponents. Factoring Algebraic Epressions Containing Fractional and Negative Eponents Although epressions containing fractional and negative eponents are not polynomials, they can be simplified using factoring techniques. EXAMPLE Factor and simplify: Factoring Involving Fractional and Negative Eponents The greatest common factor is + with the smallest eponent in the two terms. Thus, the greatest common factor is ( + ) 4. ( + ) ( + ) 4 ( + ) ( + )  4 ( + ) ( + )  4 [ + ( + )] ( + ) ( + ) 4. Epress each term as the product of the greatest common factor and its other factor. Factor out the greatest common factor. + ( + ) 4 b n b n
10 BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page 57 Eercise Set P.5 57 Factor and simplify: (  )  + (  ). EXERCISE SET P.5 Practice Eercises In Eercises 0, factor out the greatest common factor ( + 5) + ( + 5) 8. ( + ) + 4( + ) 9. (  ) + (  ) 0. ( + 5) + 7( + 5) In Eercises 6, factor by grouping In Eercises 7 0, factor each trinomial, or state that the trinomial is prime In Eercises 40, factor the difference of two squares y y In Eercises 4 48, factor any perfect square trinomials, or state that the polynomial is prime In Eercises 49 56, factor using the formula for the sum or difference of two cubes In Eercises 57 84, factor completely, or state that the polynomial is prime y 58y 74. y 56y 75. 0y 445y y4  y y y 79. 9b  6y b y 80. 6a  5y a y 8. y  6y y  7y a a In Eercises 85 94, factor and simplify each algebraic epression ( + )  ( + ) ( + 4) + ( + 4) 7 ( + 5)   ( + 5) ( + )  + ( + ) 5 (4  )  (4  ) 8(4 + )  + 0(5 + )(4 + )  Application Eercises 95. Your computer store is having an incredible sale. The price on one model is reduced by 40%.Then the sale price is reduced by another 40%. If is the computer s original price, the sale price can be represented by (  0.4)  0.4(  0.4). a. Factor out (  0.4) from each term. Then simplify the resulting epression. b. Use the simplified epression from part (a) to answer these questions: With a 40% reduction followed by a 40% reduction, is the computer selling at 0% of its original price? If not, at what percentage of the original price is it selling?
11 BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page Chapter P Prerequisites: Fundamental Concepts of Algebra 96. The polynomial describes the number, in thousands, of high school graduates in the United States years after 99. a. According to this polynomial, how many students will graduate from U.S. high schools in 00? b. Factor the polynomial. c. Use the factored form of the polynomial in part (b) to find the number of high school graduates in 00. Do you get the same answer as you did in part (a)? If so, does this prove that your factorization is correct? Eplain. 97. A rock is dropped from the top of a 56foot cliff. The height, in feet, of the rock above the water after t seconds is described by the polynomial 566t. Factor this epression completely. 98. The amount of sheet metal needed to manufacture a cylindrical tin can, that is, its surface area, S, is S r + rh. Epress the surface area, S, in factored form. In Eercises 99 00, find the formula for the area of the shaded region and epress it in factored form Writing in Mathematics 0. Using an eample, eplain how to factor out the greatest common factor of a polynomial. r h 0. Suppose that a polynomial contains four terms. Eplain how to use factoring by grouping to factor the polynomial. 0. Eplain how to factor Eplain how to factor the difference of two squares. Provide an eample with your eplanation. 05. What is a perfect square trinomial and how is it factored? 06. Eplain how to factor What does it mean to factor completely? Critical Thinking Eercises 08. Which one of the following is true? a. Because + is irreducible over the integers, it follows that + is also irreducible. b. One correct factored form for is (  4) +. c (  4) d. None of the above is true. In Eercises 09, factor completely. 09. n + 6 n y 4  y + y. (  5)  ( + 5)   ( + 5) (  5)  In Eercises 4, find all integers b so that the trinomial can be factored.. + b b Group Eercise 5. Without looking at any factoring problems in the book, create five factoring problems. Make sure that some of your problems require at least two factoring techniques. Net, echange problems with another person in your group. Work to factor your partner s problems. Evaluate the problems as you work: Are they too easy? Too difficult? Can the polynomials really be factored? Share your response with the person who wrote the problems. Finally, grade each other s work in factoring the polynomials. Each factoring problem is worth 0 points. You may award partial credit. If you take off points, eplain why points are deducted and how you decided to take off a particular number of points for the error(s) that you found.
12 BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page 59 Section P.6 Rational Epressions 59 SECTION P.6 Rational Epressions Objectives. Specify numbers that must be ecluded from the domain of rational epressions.. Simplify rational epressions.. Multiply rational epressions. 4. Divide rational epressions. 5. Add and subtract rational epressions. 6. Simplify comple rational epressions. Discovery What happens if you try substituting 00 for in 50? 00  What does this tell you about the cost of cleaning up all of the river s pollutants? Specify numbers that must be ecluded from the domain of rational epressions. How do we describe the costs of reducing environmental pollution? We often use algebraic epressions involving quotients of polynomials. For eample, the algebraic epression describes the cost, in millions of dollars, to remove percent of the pollutants that are discharged into a river. Removing a modest percentage of pollutants, say 40%, is far less costly than removing a substantially greater percentage, such as 95%. We see this by evaluating the algebraic epression for 40 and Evaluating for 0040: Cost is 95: 50(40) L Cost is 50(95) The cost increases from approimately $67 million to a possibly prohibitive $4750 million, or $4.75 billion. Costs spiral upward as the percentage of removed pollutants increases. Many algebraic epressions that describe costs of environmental projects are eamples of rational epressions. First we will define rational epressions. Then we will review how to perform operations with such epressions. Rational Epressions A rational epression is the quotient of two polynomials. Some eamples are , 4 4, ,  and The set of real numbers for which an algebraic epression is defined is the domain of the epression. Because rational epressions indicate division and division by zero is undefined, we must eclude numbers from a rational epression s domain that make the denominator zero.
13 BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page Chapter P Prerequisites: Fundamental Concepts of Algebra EXAMPLE Ecluding Numbers from the Domain Find all the numbers that must be ecluded from the domain of each rational epression: 4 a. b To determine the numbers that must be ecluded from each domain, eamine the denominators. 4 a. b.   ( + )(  ) This denominator would equal zero if. This factor would equal zero if. This factor would equal zero if. For the rational epression in part (a), we must eclude from the domain. For the rational epression in part (b), we must eclude both  and from the domain. These ecluded numbers are often written to the right of a rational epression. 4 , Z , Z, Z Find all the numbers that must be ecluded from the domain of each rational epression: 7 a. b Simplify rational epressions. Simplifying Rational Epressions A rational epression is simplified if its numerator and denominator have no common factors other than or . The following procedure can be used to simplify rational epressions: Simplifying Rational Epressions. Factor the numerator and denominator completely.. Divide both the numerator and denominator by the common factors. EXAMPLE Simplifying Rational Epressions + Simplify: a. b
14 BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page 6 Section P.6 Rational Epressions 6 a. + + ( + ) + ( + ) +, Z Factor the numerator. Because the denominator is +, Z. Divide out the common factor, +. Denominators of need not be written because a a. b ( + 5)( + ) ( + 5)(  5) ( + 5) ( + ) ( + 5) (  5) +  5, Z5, Z 5 Factor the numerator and denominator. Because the denominator is ( + 5)(  5), Z5 and Z 5. Divide out the common factor, + 5. Simplify: + a. b Multiply rational epressions. Multiplying Rational Epressions The product of two rational epressions is the product of their numerators divided by the product of their denominators. Here is a stepbystep procedure for multiplying rational epressions: Multiplying Rational Epressions. Factor all numerators and denominators completely.. Divide numerators and denominators by common factors.. Multiply the remaining factors in the numerator and multiply the remaining factors in the denominator. EXAMPLE Multiplying Rational Epressions Multiply and simplify:
15 BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page 6 6 Chapter P Prerequisites: Fundamental Concepts of Algebra ( + )(  )  (  7) ( + ) (  ) (  7) +, Z, Z 7 This is the given multiplication problem. Factor all numerators and denominators. Because the denominator has factors of  and  7, Z and Z 7. Divide numerators and denominators by common factors. Multiply the remaining factors in the numerator and denominator. 4 Divide rational epressions. Multiply and simplify: Dividing Rational Epressions We find the quotient of two rational epressions by inverting the divisor and multiplying. EXAMPLE 4 Divide and simplify: (  4)( + ) ( + )(  ) These ecluded numbers from the domain must also be ecluded from the simplified epression s domain. (  4) ( + ) ( + ) (  ) Dividing Rational Epressions, ( + ) (  4) +, Z, Z, Z , This is the given division problem. Invert the divisor and multiply. Factor throughout. For nonzero denominators, Z, Z, and Z 4. Divide numerators and denominators by common factors. Multiply the remaining factors in the numerator and the denominator. 4 Divide and simplify:  + +,
16 BLITMCPB.QXP.0599_4874_tg //0 :8 PM Page 6 5 Add and subtract rational epressions. Section P.6 Rational Epressions 6 Adding and Subtracting Rational Epressions with the Same Denominator We add or subtract rational epressions with the same denominator by () adding or subtracting the numerators, () placing this result over the common denominator, and () simplifying, if possible. Study Tip Eample 5 shows that when a numerator is being subtracted, we must subtract every term in that epression. EXAMPLE 5 Subtract: Subtracting Rational Epressions with the Same Denominator (4  ) ( + ) (  ) , Z, Z Subtract numerators and include parentheses to indicate that both terms are subtracted. Place this difference over the common denominator. Remove parentheses and then change the sign of each term. Combine like terms. Factor and simplify ( Z and Z ). 5 Subtract: Adding and Subtracting Rational Epressions with Different Denominators Rational epressions that have no common factors in their denominators can be added or subtracted using one of the following properties: The denominator, bd, is the product of the factors in the two denominators. Because we are looking at rational epressions that have no common factors in their denominators, the product bd gives the least common denominator. EXAMPLE 6 Subtract: a b + c d ad + bc bd a b  c d Subtracting Rational Epressions Having No Common Factors in Their Denominators ad  bc, b Z 0, d Z 0. bd We need to find the least common denominator. This is the product of the distinct factors in each denominator, namely (  )( + ). We can therefore use the subtraction property given previously as follows:
17 BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page Chapter P Prerequisites: Fundamental Concepts of Algebra a b c d ad bc bd ( + )( + )  (  )4 (  )( + ) (8  ) (  )( + ) (  )( + ) (  )( + ), Z, Z Observe that a +, b , c 4, and d +. Multiply. Remove parentheses and then change the sign of each term. Combine like terms in the numerator. 6 Add: The least common denominator, or LCD, of several rational epressions is a polynomial consisting of the product of all prime factors in the denominators, with each factor raised to the greatest power of its occurrence in any denominator. When adding and subtracting rational epressions that have different denominators with one or more common factors in the denominators, it is efficient to find the least common denominator first. Finding the Least Common Denominator. Factor each denominator completely.. List the factors of the first denominator.. Add to the list in step any factors of the second denominator that do not appear in the list. 4. Form the product of each different factor from the list in step. This product is the least common denominator. EXAMPLE 7 Find the least common denominator of Finding the Least Common Denominator and Step Factor each denominator completely. Step ( + ) ( + ) List the factors of the first denominator. 5,, ( + )
18 BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page 65 Section P.6 Rational Epressions 65 Step Add any unlisted factors from the second denominator. The second denominator is ( + ) or ( + )( + ). One factor of + is already in our list, but the other factor is not. We add + to the list. We have 5,, ( + ), ( + ). Step 4 The least common denominator is the product of all factors in the final list. Thus, 5( + )( + ), is the least common denominator. or 5( + ) 7 Find the least common denominator of and 79. Finding the least common denominator for two (or more) rational epressions is the first step needed to add or subtract the epressions. Adding and Subtracting Rational Epressions That Have Different Denominators with Shared Factors. Find the least common denominator.. Write all rational epressions in terms of the least common denominator. To do so, multiply both the numerator and the denominator of each rational epression by any factor(s) needed to convert the denominator into the least common denominator.. Add or subtract the numerators, placing the resulting epression over the least common denominator. 4. If necessary, simplify the resulting rational epression. EXAMPLE 8 Add: Adding Rational Epressions with Different Denominators Step Find the least common denominator. Start by factoring the denominators. +  ( + )(  )  ( + )(  ) The factors of the first denominator are + and . The only factor from the second denominator that is not listed is +. Thus, the least common denominator is ( + )(  )( + ).
19 BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page Chapter P Prerequisites: Fundamental Concepts of Algebra Step Write all rational epressions in terms of the least common denominator. We do so by multiplying both the numerator and the denominator by any factor(s) needed to convert the denominator into the least common denominator ( + )(  ) + ( + )(  ) ( + )( + ) + ( + ) ( + )(  )( + ) ( + )(  )( + ) , Z, Z, Z ( + )(  )( + ) The least common denominator is ( + )(  )( + ). ( + )( + ) ( + )(  )( + ) + ( + ) Multiply each numerator and ( + )(  )( + ) denominator by the etra factor required to form ( + )(  )( + ), the least common denominator. Step Add numerators, putting this sum over the least common denominator. Perform the multiplications in the numerator. Combine like terms in the numerator. Step 4 If necessary, simplify. Because the numerator is prime, no further simplification is possible. 8 Subtract: Simplify comple rational epressions. Comple Rational Epressions Comple rational epressions, also called comple fractions, have numerators or denominators containing one or more rational epressions. Here are two eamples of such epressions: +  Separate rational epressions occur in the numerator and denominator. + h . h Separate rational epressions occur in the numerator. One method for simplifying a comple rational epression is to combine its numerator into a single epression and combine its denominator into a single epression. Then perform the division by inverting the denominator and multiplying.
20 BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page 67 Section P.6 Rational Epressions 67 EXAMPLE 9 Simplifying a Comple Rational Epression Simplify: , Z , , Z 0, Z The terms in the numerator and in the denominator are each combined by performing the addition and subtraction. The least common denominator is. Perform the addition in the numerator and the subtraction in the denominator. Rewrite the main fraction bar as,. Invert the divisor and multiply ( Z 0 and Z ). Divide a numerator and denominator by the common factor,. Multiply the remaining factors in the numerator and in the denominator. 9 Simplify: A second method for simplifying a comple rational epression is to find the least common denominator of all the rational epressions in its numerator and denominator. Then multiply each term in its numerator and denominator by this least common denominator. Here we use this method to simplify the comple rational epression in Eample a + b a  b The least common denominator of all the rational epressions is. Multiply the numerator and denominator by. Because we are not, changing the comple fraction ( Z 0). +  Use the distributive property. Be sure to distribute to every term. + , Z 0, Z Multiply. The comple rational epression is now simplified.
21 BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page Chapter P Prerequisites: Fundamental Concepts of Algebra EXERCISE SET P.6 Practice Eercises In Eercises 6, find all numbers that must be ecluded from the domain of each rational epression In Eercises 7 4, simplify each rational epression. Find all numbers that must be ecluded from the domain of the simplified rational epression y + 7y  8 y  4y y  y + y + 5y In Eercises 5, multiply or divide as indicated , , , , , , , In Eercises 54, add or subtract as indicated , , ,
22 BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page 69 Eercise Set P.6 69 In Eercise 55 64, simplify each comple rational epression Anthropologists and forensic scientists classify skulls using L + 60W L  L  40W L where L is the skull s length and W is its width. + y y  y L W Application Eercises 65. The rational epression describes the cost, in millions of dollars, to inoculate percent of the population against a particular strain of flu. a. Evaluate the epression for 40, 80, and 90. Describe the meaning of each evaluation in terms of percentage inoculated and cost. b. For what value of is the epression undefined? c. What happens to the cost as approaches 00%? How can you interpret this observation? 66. Doctors use the rational epression DA A to determine the dosage of a drug prescribed for children. In this epression, A child s age, and D adult dosage. What is the difference in the child s dosage for a 7yearold child and a yearold child? Epress the answer as a single rational epression in terms of D.Then describe what your answer means in terms of the variables in the rational epression. a. Epress the classification as a single rational epression. b. If the value of the rational epression in part (a) is less than 75, a skull is classified as long. A medium skull has a value between 75 and 80, and a round skull has a value over 80. Use your rational epression from part (a) to classify a skull that is 5 inches wide and 6 inches long. 68. The polynomial 6t 407t + 8t  66t + 5,0 describes the annual number of drug convictions in the United States t years after 984. The polynomial 8t 47t + 596t  695t + 7,44 describes the annual number of drug arrests in the United States t years after 984. Write a rational epression that describes the conviction rate for drug arrests in the United States t years after The average speed on a roundtrip commute having a oneway distance d is given by the comple rational epression r d d + d r r in which r and are the speeds on the outgoing and return trips, respectively. Simplify the epression. Then find the average speed for a person who drives from home to work at 0 miles per hour and returns on the same route averaging 0 miles per hour. Eplain why the answer is not 5 miles per hour.
23 BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page Chapter P Prerequisites: Fundamental Concepts of Algebra Writing in Mathematics 70. What is a rational epression? 7. Eplain how to determine which numbers must be ecluded from the domain of a rational epression. 7. Eplain how to simplify a rational epression. 7. Eplain how to multiply rational epressions. 74. Eplain how to divide rational epressions. 75. Eplain how to add or subtract rational epressions with the same denominators. 76. Eplain how to add rational epressions having no common factors in their denominators. Use in your eplanation Eplain how to find the least common denominator for denominators of  00 and Describe two ways to simplify +. Eplain the error in Eercises Then rewrite the right side of the equation to correct the error that now eists a + b a + b 8. a + a b a + b 8. A politician claims that each year the conviction rate for drug arrests in the United States is increasing. Eplain how to use the polynomials in Eercise 68 to verify this claim. Technology Eercises 8. How much are your monthly payments on a loan? If P is the principal, or amount borrowed, i is the monthly interest rate, and n is the number of monthly payments, then the amount, A, of each monthly payment is A  Pi ( + i) n a. Simplify the comple rational epression for the amount of each payment. b. You purchase a $0,000 automobile at % monthly interest to be paid over 48 months. How much do you pay each month? Use the simplified rational epression from part (a) and a calculator. Round to the nearest dollar. Critical Thinking Eercises 84. Which one of the following is true?  5 a b. if Z 0 and y Z 0. y, y, c. The least common denominator needed to find is d. The rational epression is not defined for 4. However, as gets closer and closer to 4, the value of the epression approaches 8. In Eercises 85 86, find the missing epression In one short sentence, five words or less, eplain what does to each number (  )( + )
24 BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page 7 Chapter Summary, Review, and Test 7 CHAPTER SUMMARY, REVIEW, AND TEST Summary: Basic Formulas Definition of Absolute Value b if 0  if 6 0 Distance between s a and b on a Number Line Properties of Algebra Commutative Associative Distributive Identity Inverse a  b or b  a a + b b + a, ab ba (a + b) + c a + (b + c) (ab)c a(bc) a(b + c) ab + ac a + 0 a a + (a) 0 a a a a Z 0 a, Product and Quotient Rules for nth Roots Rational Eponents Special Products n a n b n ab a n n a, a m n A n ab m n a m, (A + B)(A  B) A  B n a n b n a Bb a  n a n n a, a  m n a m n (A + B) A + AB + B (A  B) A  AB + B (A + B) A + A B + AB + B (A  B) A  A B + AB  B Properties of Eponents (b m ) n b mn, b  n b n, b m b n bm  n, b 0, b m b n b m + n, (ab) n a n b n, a a b b n an b n Factoring Formulas A  B (A + B)(A  B) A + AB + B (A + B) A  AB + B (A  B) A + B (A + B) (A  AB + B ) A  B (A  B) (A + AB + B ) Review Eercises You can use these review eercises, like the review eercises at the end of each chapter, to test your understanding of the chapter s topics. However, you can also use these eercises as a prerequisite test to check your mastery of the fundamental algebra skills needed in this book. P.. Consider the set: E7,  9, 0, 0.75,, p, 8F. List all numbers from the set that are a. natural numbers, b. whole numbers, c. integers, d. rational numbers, e. irrational In Eercises 4, rewrite each epression without absolute value bars Epress the distance between the numbers 7 and 4 using absolute value. Then evaluate the absolute value.
25 BLITMCPB.QXP.0599_4874_tg //0 :9 PM Page 7 7 Chapter P Prerequisites: Fundamental Concepts of Algebra In Eercises 6 7, evaluate each algebraic epression for the given value of the variable. 5 8( + 5) 6. (F  ); F , In Eercises 8, state the name of the property illustrated (6 ) 9 6 ( 9) In Eercises 4 5, simplify each algebraic epression. 4. (75y)  (4y  + ) 5. P. Evaluate each eponential epression in Eercises () () Simplify each eponential epression in Eercises ( 4 y ). (5 y ) (  y  ). ( ) y y  In Eercises 4 5, write each number in decimal notation * * 05 In Eercises 6 7, write each number in scientific notation. 6.,590, In Eercises 8 9, perform the indicated operation and write the answer in decimal notation. 6.9 * 0 8. ( * 0 )(. * 0 ) 9. * If you earned $ million per year ($0 6 ), how long would it take to accumulate $ billion ($0 9 )?. If the population of the United States is.8 * 0 8 and each person spends about $50 per year going to the movies (or renting movies), epress the total annual spending on movies in scientific notation. P. A 5 + B 5 + (6 9) (6 9) A 5 + B A 5 + B ( 7) + (4 7) (4 7) + ( 7) 5 (5) + C(y) + (y)d  () Use the product rule to simplify the epressions in Eercises 5. In Eercises 4 5, assume that variables represent nonnegative real numbers r Use the quotient rule to simplify the epressions in Eercises B 4 96 (Assume that 7 0.) In Eercises 8 40, add or subtract terms whenever possible In Eercises 4 44, rationalize the denominator Evaluate each epression in Eercises or indicate that the root is not a real number (5) 4 Simplify the radical epressions in Eercises y (Assume that 7 0. ) 4 6 In Eercises 54 59, evaluate each epression In Eercises 60 6, simplify using properties of eponents. 60. (5 ) (4 4 ) Simplify by reducing the inde of the radical: 6 y. P (5 6 ) In Eercises 64 65, perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree. 64. ( ) + ( )
26 BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page 7 In Eercises 75 79, find each product. 75. ( + 7y)(  5y) 76. (  5y) 77. ( + y) 78. (7 + 4y)(74y) 79. (a  b)(a + ab + b ) Chapter Summary, Review, and Test ( )  ( ) In Eercises 97 99, factor and simplify each algebraic epression. In Eercises 66 7, find each product (  )( ) 67. (  5)( + ) 98. (  4)( + )  (  4) ( + ) 68. (4 + 5)(45) 69. ( + 5) (  4) 7. ( + ) 7. (5  ) P.6 In Eercises 00 0, simplify each rational epression. Also, In Eercises 7 74, perform the indicated operations. Indicate the degree of the resulting polynomial. list all numbers that must be ecluded from the domain. 7. (78y + y ) + (89y  4y ) ( y  5 y  9 )  ( y  6 y +  4) In Eercises 0 05, multiply or divide as indicated , P.5 In Eercises 80 96, factor completely, or state that the polynomial is prime y y , In Eercises 06 09, add or subtract as indicated In Eercises 0, simplify each comple rational epression Chapter P Test. List all the rational numbers in this set: 0 y 4 E7,  4 5, 0, 0.5,, 4, 7, pf y 4 In Eercises, state the name of the property illustrated. 7. 6r r (Assume that r 0.). ( + 5) (5 + ). 6(7 + 4) Epress in scientific notation: Simplify each epression in Eercises (0  y)  5(  4y + ) Evaluate: 75.
27 BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page Chapter P Prerequisites: Fundamental Concepts of Algebra In Eercises 4, find each product.. (  5)( ) 4. (5 + y) In Eercises 5 0, factor completely, or state that the polynomial is prime y y. Factor and simplify: ( + ) ( + ) 5. In Eercises 5, perform the operations and simplify, if possible ,
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