Chapter 5 POLYNOMIALS: FACTORING
Name:Instructor:Date:Section:Chapter 5 POLYNOMIALS: FACTORING5.1 Introduction to FactoringLearning Objectivesa Find the greatest common factor, the GCF, of monomials.b Factor polynomials when the terms have a common factor, factoring out the greatestcommon factor.c Factor certain expressions with four terms using factoring by grouping.Key TermsUse the vocabulary terms listed below to complete each statement in Exercises 1–4. Termsmay be used more than once.factorfactoring by groupingfactorization1.To a polynomial is to express it as a product.2.A(n) of a polynomial P is a polynomial that can be used toexpress P as a product.3.A(n) of a polynomial is an expression that names thatpolynomial as a product.4.Certain polynomials with four terms can be factored using .Objective aFind the greatest common factor, the GCF, of monomials.Find the GCF.5.x 3 , 10 x5.6.4 x5 , x36.7.3x, 3x 5 , 127.167Copyright 2011 Pearson Education, Inc. Publishing as Addison-Wesley.
8. 11x 4 y 3 , 33x 3 y 5 , 55 x 2 y8.9. 15 x, 13x 2 , 18 x 79.10. p 3q3 , p 7 q2 , p 4 q6 , p 5q810.Objective b Factor polynomials when the terms have a common factor, factoring outthe greatest common factor.Factor. Check by multiplying.11. x 2 10 x11.12. 3 x 6 12 x 412.13. 10 x 5 5 x 213.14. 4 x 2 4 x 2014.15. 11x 4 y 3 33x3 y 5 55 x 2 y15.168Copyright 2011 Pearson Education, Inc. Publishing as Addison-Wesley.
Name:Instructor:Date:Section:16. 8 x5 10 x 4 7 x316.17. p3 q 4 p 3 q 2 p 4 q3 p 6 q817.18. 3 x8 30 x 6 15 x5 18 x318.19. 1.4 x 4 3.5 x3 4.2 x 2 7.0 x19.3 7 7 5 1 3 1 2x x x x222220.20.Objective cFactor certain expressions with four terms using factoring by grouping.Factor.21. x3 ( x 1) 3 ( x 1)21.22. 7 p 2 ( 4 p 5 ) ( 4 p 5 )22.169Copyright 2011 Pearson Education, Inc. Publishing as Addison-Wesley.
Factor by grouping.23. x3 2 x 2 5 x 1023.24. 2 z 3 2 z 2 3 z 324.25. 8 p 3 20 p 2 10 p 2525.26. 20 x 3 15 x 2 8 x 626.27. 3 x 3 6 x 2 x 227.28. x3 3 x 2 7 x 2128.170Copyright 2011 Pearson Education, Inc. Publishing as Addison-Wesley.
Name:Instructor:Date:Section:Chapter 5 POLYNOMIALS: FACTORING5.2 Factoring Trinomials of the Type x 2 bx cLearning Objectivea Factor trinomials of the type x 2 bx c by examining the constant term c .Key TermsUse the vocabulary terms listed below to complete each statement in Exercises 1–4.leading coefficientnegativepositiveprime1.In the expression ax 2 bx c, a is called the .2.A(n) polynomial cannot be factored further.3.When the constant term of a trinomial is , the constant terms ofthe binomial factors have the same sign.4.When the constant term of a trinomial is , the constant terms ofthe binomial factors have opposite signs.Objective aterm c .Factor trinomials of the type x 2 bx c by examining the constantFactor. Remember that you can check by multiplying.5.5.x 2 9 x 18Pairs ofFactorsSums ofFactors171Copyright 2011 Pearson Education, Inc. Publishing as Addison-Wesley.
6.7.8.9.x 2 8 x 16Pairs ofFactorsa 2 a 20Pairs ofFactorsy2 9 y 8Pairs ofFactorst 2 13t 306.Sums ofFactors7.Sums ofFactors8.Sums ofFactors9.10. p 2 5 p 1410.11. x 2 3 x 111.172Copyright 2011 Pearson Education, Inc. Publishing as Addison-Wesley.
Name:Instructor:Date:Section:12. x3 5 x 2 6 x12.13. x3 3 x 2 40 x13.14. 16 x 60 x 214.15. t 4 7t 2 1015.16. 15 5 p p 216.17. x 2 22 x 12117.18. 27 6a a 218.19. x 4 13 x3 30 x 219.173Copyright 2011 Pearson Education, Inc. Publishing as Addison-Wesley.
20. y 2 16 y 6020.21. 75 28t t 221.22. x 2 0.1x 0.0622.23. 60 7t t 223.24. p 2 pq 20q 224.25. m 2 5mn 66n 225.26. 5a 8 15a 7 140a 626.174Copyright 2011 Pearson Education, Inc. Publishing as Addison-Wesley.
Name:Instructor:Date:Section:Chapter 5 POLYNOMIALS: FACTORING5.3 Factoring ax 2 bx c , a 1 : The FOIL MethodLearning Objectivea Factor trinomials of the type ax 2 bx c, a 1 , using the FOIL method.Key TermsUse the vocabulary terms listed below to complete each statement in Exercises 1–4.FirstInsideLastOutside1.In the product ( x 3)( 2 x 1) , x and 2x are the terms.2.In the product ( x 3)( 2 x 1) , 3 and 1 are the terms.3.In the product ( x 3)( 2 x 1) , 3 and 2 x are the terms.4.In the product ( x 3)( 2 x 1) , x and 1 are the terms.Objective aFactor trinomials of the type ax 2 bx c , a 1 , using the FOIL method.Factor.5.2 x 2 5 x 125.6.5 x 2 32 x 126.7.6 x 2 17 x 57.175Copyright 2011 Pearson Education, Inc. Publishing as Addison-Wesley.
8.3x 2 2 x 18.9.16 x 2 8 x 159.10. 18 x 2 15 x 210.11. 21 58 x 21x 211.12. 25 x 2 40 x 1612.13. 7 5 x 2 x 213.14. 8 x 2 28 x 3614.176Copyright 2011 Pearson Education, Inc. Publishing as Addison-Wesley.
Name:Instructor:Date:Section:15. 15 x 2 55 x 5015.16. 1 3 x 2 2 x16.17. 16 x 2 72 x 4017.18. 21t 6t 2 918.19. 10 p 4 21 p3 10 p 219.20. 14a 4 23a 2 320.177Copyright 2011 Pearson Education, Inc. Publishing as Addison-Wesley.
21. 81 y 2 36 y 421.22. 16 x 2 21x 2522.23. 6m 2 11mn 10n 223.24. 20a 2 29ab 6b 224.25. 14 p 2 23 pq 3q 225.26. 16 x 2 8 xy 8 y 226.178Copyright 2011 Pearson Education, Inc. Publishing as Addison-Wesley.
Name:Instructor:Date:Section:Chapter 5 POLYNOMIALS: FACTORING5.4 Factoring ax 2 bx c , a 1 : The ac-MethodLearning Objectivea Factor trinomials of the type ax 2 bx c, a 1 , using the ac-method.Key TermsUse the vocabulary terms listed below to complete the steps for factoring using theac-method in Exercises 1–6.common factorgroupingleading coefficientmultiplyingsplitsum1.Factor out a(n) , if any.2.Multiply the a and the constant c.3.Try to factor the product ac so that the of the factors is b.4.the middle term.5.Factor by .6.Check by .Objective aFactor trinomials of the type ax 2 bx c , a 1 , using the ac-method.Factor. Note that the middle term has already been split.7.x 2 4 x 3 x 127.8.6 x 2 4 x 15 x 108.179Copyright 2011 Pearson Education, Inc. Publishing as Addison-Wesley.
9.24 y 2 20 y 6 y 510. 2 x 4 4 x 2 7 x 2 149.10.Factor using the ac-method.11. 3 x 2 13 x 1011.12. 8 x 2 26 x 1512.13. 4 x 2 8 x 2113.14. 4 x 2 16 x 914.15. 14 x 2 65 x 915.16. 25a 2 30a 916.180Copyright 2011 Pearson Education, Inc. Publishing as Addison-Wesley.
Name:Instructor:Date:Section:17. 3 5a 2a 217.18. 15 x 2 99 x 4218.19. 15t 2 22t 919.20. 4 x 2 7 x 220.21. 6t 3 19t 2 15t21.22. 26 y 6 y 2 822.23. 12 p 4 4 p 3 5 p 223.24. 105 x3 50 x 2 5 x24.181Copyright 2011 Pearson Education, Inc. Publishing as Addison-Wesley.
25. 8 x 4 26 x 2 1525.26. 16 x 2 72 x 8126.27. 15 y 2 10 y 1827.28. 15a 2 11ab 2b 228.29. 6 s 2 20 st 14t 229.30. 8 x 2 28 xy 24 y 230.182Copyright 2011 Pearson Education, Inc. Publishing as Addison-Wesley.
Name:Instructor:Date:Section:Chapter 5 POLYNOMIALS: FACTORING5.5 Factoring Trinomial Squares and Differences of SquaresLearning Objectivesa Recognize trinomial squares.b Factor trinomial squares.c Recognize differences of squares.d Factor differences of squares, being careful to factor completely.Key TermsUse the vocabulary terms listed below to complete each statement in Exercises 1–4.difference of squaresfactored completelysum of squarestrinomial square1.The expression x 2 10 x 25 is a(n) .2.The expression 9 x 2 16 is a(n) .3.The expression y 2 81 is a(n) .4.When no factor can be factored further, we have .Objective aRecognize trinomial squares.Determine whether each of the following is a trinomial square.5.x 2 10 x 255.6.4 x 2 12 x 96.183Copyright 2011 Pearson Education, Inc. Publishing as Addison-Wesley.
Objective bFactor trinomial squares.Factor completely. Remember to look first for a common factor and to check bymultiplying.7.x2 6x 97.8.36 12 x x 28.9.m 4 10m 2 259.10. 12a 2 84a 14710.11. x3 20 x 2 100 x11.12. 64 80 x 25 x 212.13. 4m 2 20mn 25n 213.14. 4 p 4 16 p 2 1614.15. x 2 8 xy 16 y 215.16. 81a 2 18ab b 216.184Copyright 2011 Pearson Education, Inc. Publishing as Addison-Wesley.
Name:Instructor:Date:Section:Objective cRecognize differences of squares.Determine whether each of the following is a difference of squares.17. x 2 10017.18. 4 x 2 10 y 218.19. x 2 1617.Objective dFactor differences of squares, being careful to factor completely.Factor completely. Remember to look first for a common factor.20. a 2 8120.21. 100 y 221.22. 64x 2 y 222.185Copyright 2011 Pearson Education, Inc. Publishing as Addison-Wesley.
23. 4m 2 49n 223.24. 18 x 2 3224.25. 0.04a 2 0.002525.26. 36 p 4 126.27. 3 x 4 4827.28. x16 8128.29. 9 1 2x1629.186Copyright 2011 Pearson Education, Inc. Publishing as Addison-Wesley.
Name:Instructor:Date:Section:Chapter 5 POLYNOMIALS: FACTORING5.6 Factoring: A General StrategyLearning Objectivea Factor polynomials completely using any of the methods considered in this chapter.Key TermsUse the vocabulary terms listed below to complete each statement in Exercises 1–4.completely1.differencegroupingsquareWhen factoring a polynomial with two terms, determine whether you havea(n) of squares.2.When factoring a polynomial with three terms, determine whether the trinomial isa(n) .3.When factoring a polynomial with four terms, try factoring by .4.Always factor .Objective a Factor polynomials completely using any of the methods considered inthis chapter.Factor completely.5.5t 2 205.6.x 2 18 x 816.187Copyright 2011 Pearson Education, Inc. Publishing as Addison-Wesley.
7.6 x 2 11x 107.8.x3 3x 2 x 38.9.x 4 5 x 2 4 x3 20 x9.10. 6 x3 4 x 2 10 x10.11. n 2 111.12. 24a 2 612.13. 2 x 4 12 x3 18 x 213.14. x 2 3 x 714.15. 25 45 x 10 x 215.188Copyright 2011 Pearson Education, Inc. Publishing as Addison-Wesley.
Name:Instructor:Date:Section:16. 3 x 5 3 x16.17. 3 p 3 q 12 pq 317.18. 4 x ( u 2 3v ) ( u 2 3v )18.19. a 2 a ab b19.20. 20m3 n5 5m 2 n 3 15m 4 n 420.21. 9c 2 4d 2 12cd21.22. 25 x 2 z 2 40 xyz 16 y 222.23. m 2 4mn 60n 223.189Copyright 2011 Pearson Education, Inc. Publishing as Addison-Wesley.
24. a 2b 2 2ab 1524.25. x5 y 2 4 x 4 y 32 x325.126. r 2 t 2426.27. c8 d 4 8127.28. 49m4 70m3 n 25m2 n 228.190Copyright 2011 Pearson Education, Inc. Publishing as Addison-Wesley.
Name:Instructor:Date:Section:Chapter 5 POLYNOMIALS: FACTORING5.7 Solving Quadratic Equations by FactoringLearning Objectivesa Solve equations (already factored) using the principle of zero products.b Solve equations by factoring and then using the principle of zero products.Key TermsUse the vocabulary terms listed below to complete each statement in Exercises 1–4.factorproductsquadratic equationzero1.3 x 2 5 x 3 0 is an example of a(n) .2.The principle of zero states that a product is 0 if and only if oneor both of the factors is 0.3.To use the principle of zero products, you must have on one sideof the equation.4.To use the principle of zero products, we a quadraticpolynomial.Objective a Solve equations (already factored) using the principle of zero products.Solve using the principle of zero products.5.( x 3)( x 10 ) 05.6.( x 5)( x 7 ) 06.191Copyright 2011 Pearson Education, Inc. Publishing as Addison-Wesley.
7.x ( x 15 ) 07.8.0 x ( x 8)8.9.( 4 x 3)( 2 x 10 ) 09.10.( 6 x 11)(8 x 3) 010.11.12.1 3x21 4x 03( 0.1x 0.2 )( 0.5 x 15) 011.12.192Copyright 2011 Pearson Education, Inc. Publishing as Addison-Wesley.
Name:Instructor:Objective bproducts.Date:Section:Solve equations by factoring and then using the principle of zeroSolve by factoring and using the principle of zero products. Remember to check.13. x 2 5 x 4 013.14. x 2 4 x 21 014.15. x 2 10 x 015.16. x 2 12 x 016.17. x 2 2517.18. 16 x 2 1 018.193Copyright 2011 Pearson Education, Inc. Publishing as Addison-Wesley.
19. 0 4 x x 2 419.20. 3 x 2 10 x20.21. 3 x 2 5 x 221.22. 20n 2 11n 322.23. 2 y 2 16 y 30 023.24. t ( 3t 5 ) 2824.194Copyright 2011 Pearson Education, Inc. Publishing as Addison-Wesley.
Name:Instructor:Date:Section:Find the x-intercepts for the graph of the equation. (The grids are intentionally not included.)25.25.26.26.195Copyright 2011 Pearson Education, Inc. Publishing as Addison-Wesley.
27. Use the following graph to solve x 2 x 2 0 .27.28. Use the following graph to solve x 2 2 x 3 0 .28.196Copyright 2011 Pearson Education, Inc. Publishing as Addison-Wesley.
Name:Instructor:Date:Section:Chapter 5 POLYNOMIALS: FACTORING5.8 Applications of Quadratic EquationsLearning Objectivesa Solve applied problems involving quadratic equations that can be solved by factoring.Key TermsUse the vocabulary terms listed below to complete each statement in Exercises 1–4.hypotenuselegsPythagoreanright1.A triangle that has a 90 angle is a(n) triangle.2.In a right triangle, the side opposite the 90 angle is the .3.The sides that form the 90 angle in a right triangle are called .4.The theorem states that, in a right triangle, a 2 b 2 c 2 .Objective a Solve applied problems involving quadratic equations that can be solvedby factoring.Solve.5.The length of a rectangular table is 5 ft greater than thewidth. The area of the table is 24 ft². Find the length andthe width.5.197Copyright 2011 Pearson Education, Inc. Publishing as Addison-Wesley.
6.A rectangular serving tray is twice as long as it is wide.The area of the tray is 338 in². Find the dimensions of thetray.6.7.A triangle is 8 cm wider than it is tall. The area is 42 cm².Find the height and the base.7.8.A triangular garden has a base twice as long as its height.The garden has an area of 49 m². Find the base and theheight.8.9.A soccer league has 12 teams. What is the total number ofgames to be played if each team plays every other teamtwice? Use x 2 x N .9.10. A softball league plays a total of 210 games. How many10.teams are in the league if each team plays every other teamtwice? Use x 2 x N , where N is the number of gamesand x is the number of teams.198Copyright 2011 Pearson Education, Inc. Publishing as Addison-Wesley.
Name:Instructor:Date:Section:11. There are 30 people at a party. How many handshakes are1possible? Use N ( x 2 x ) , where N is the number of2handshakes and x is the number of people.11.12. Everyone at a meeting shook hands with each other. Therewere 276 handshakes in all. How many people were at the1meeting? Use N ( x 2 x ) , where N is the number of2handshakes and x is the number of people.12.13. During a toast at a party, there were 66 “clinks” of glasses. 13.How many people took part in the toast? Use1N x 2 x , where N is the number of clinks and x is2the number of people.()14. The product of the page numbers on two facing pages of abook is 110. Find the page numbers.14.15. The product of two consecutive even integers is 288. Findthe integers.15.16. The product of two consecutive odd integers is 195. Findthe integers.16.199Copyright 2011 Pearson Education, Inc. Publishing as Addison-Wesley.
17. The length of one leg of a right triangle is 15 ft. The length 17.of the hypotenuse is 5 ft longer than the other leg. Find thelength of the hypotenuse and the other leg.18. A metal sign is a right triangle that is 30 in. tall and has ahypotenuse of length 34 in. If the metal costs 0.10 persquare inch, find the cost of the sign.18.19. Johann landed his Cessna 172, traveling 13,000 ft over a12,000-ft horizontal distance. From what altitude did thedescent begin?19.20. The guy wire on a tower is 10 ft longer than the height ofthe tower. If the guy wire is anchored 50 ft from the footof the tower, how tall is the tower?20.21. A model rocket is launched with an initial velocity of 160ft/sec. Its height h, in feet, after t seconds is given by the21.formula h 160t 16t 2 . After how many seconds will therocket first reach a height of 336 ft?22. The sum of the squares of two consecutive odd positiveintegers is 202. Find the integers22.200Copyright 2011 Pearson Education, Inc. Publishing as Addison-Wesley.
Chapter 5 POLYNOMIALS: FACTORING 5.4 Factoring ax bx c2 , a 1: The ac-Method Learning Objective a Factor trinomials of the type ax bx c2 , a 1, using the ac-method. Key Terms Use the vocabulary terms listed below to complete the steps for factoring using the ac-method in Exercises 1–6. common factor grouping leading coefficient
Factoring . Factoring. Factoring is the reverse process of multiplication. Factoring polynomials in algebra has similar role as factoring numbers in arithmetic. Any number can be expressed as a product of prime numbers. For example, 2 3. 6 Similarly, any
Factoring Polynomials Martin-Gay, Developmental Mathematics 2 13.1 – The Greatest Common Factor 13.2 – Factoring Trinomials of the Form x2 bx c 13.3 – Factoring Trinomials of the Form ax 2 bx c 13.4 – Factoring Trinomials of the Form x2 bx c by Grouping 13.5 – Factoring Perfect Square Trinomials and Difference of Two Squares
Section 5.4 Factoring Polynomials 231 5.4 Factoring Polynomials Factoring Polynomials Work with a partner. Match each polynomial equation with the graph of its related polynomial function. Use the x-intercepts of the graph to write each polynomial in factored form. Explain your reason
Factoring . Factoring. Factoring is the reverse process of multiplication. Factoring polynomials in algebra has similar role as factoring numbers in arithmetic. Any number can be expressed as a product of prime numbers. For example, 2 3. 6 Similarly, any
Factoring Polynomials Factoring Quadratic Expressions Factoring Polynomials 1 Factoring Trinomials With a Common Factor Factoring Difference of Squares . Alignment of Accuplacer Math Topics and Developmental Math Topics with Khan Academy, the Common Core and Applied Tasks from The New England Board of Higher Education, April 2013
Section P.5 Factoring Polynomials 49 1 Factor out the greatest common factor of a polynomial. 2 Factor by grouping. 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.
Section 3.5: Multiplying Polynomials Objective: Multiply polynomials. Multiplying polynomials can take several different forms based on what we are multiplying. We will first look at multiplying monomials; then we will multiply monomials by polynomials; and finish with multiplying polynomials by polynomials.
components were orientated according to the ASTM F 1440 and fixed using a high edge retention metallographic resin to the cement indication markers given on the femoral stem. In each case, the head – neck interface was immersed in 100 mL of 0.9 g/L NaCl. The head force was actuated against
