chapter 5 polynomials: factoring - ms. wilson's...

Name: Date: Instructor: Section:

167

Chapter 5 POLYNOMIALS: FACTORING

5.1 Introduction to Factoring Learning Objectives a Find the greatest common factor, the GCF, of monomials. b Factor polynomials when the terms have a common factor, factoring out the greatest

common factor. c Factor certain expressions with four terms using factoring by grouping. Key Terms Use the vocabulary terms listed below to complete each statement in Exercises 1–4. Terms may be used more than once.

factor factoring by grouping factorization 1. To ___________________ a polynomial is to express it as a product. 2. A(n) ___________________ of a polynomial P is a polynomial that can be used to

express P as a product. 3. A(n) ___________________ of a polynomial is an expression that names that

polynomial as a product. 4. Certain polynomials with four terms can be factored using ___________________. Objective a Find the greatest common factor, the GCF, of monomials. Find the GCF. 5. 3, 10x x! 5. ________________

6. 5 34 , x x 6. ________________

7. 53 , 3 , 12x x 7. ________________

Copyright © 2011 Pearson Education, Inc. Publishing as Addison-Wesley.

168

8. 4 3 3 5 211 , 33 , 55x y x y x y! 8. ________________

9. 2 715 , 13 , 18x x x! 9. ________________

10. 3 3 7 2 4 6 5 8, , , p q p q p q p q 10. ________________

Objective b Factor polynomials when the terms have a common factor, factoring out the greatest common factor. Factor. Check by multiplying. 11. 2 10x x! 11. ________________

12. 6 43 12x x+ 12. ________________

13. 5 210 5x x! 13. ________________

14. 24 4 20x x+ ! 14. ________________

15. 4 3 3 5 211 33 55x y x y x y! ! 15. ________________



169

16. 5 4 38 10 7x x x+ ! 16. ________________

17. 3 4 3 2 4 3 6 8p q p q p q p q+ ! ! 17. ________________

18. 8 6 5 33 30 15 18x x x x! + + 18. ________________

19. 4 3 21.4 3.5 4.2 7.0x x x x! + + 19. ________________

20. 7 5 3 23 7 1 12 2 2 2

x x x x+ ! + 20. ________________

Objective c Factor certain expressions with four terms using factoring by grouping. Factor. 21. ( ) ( )3 1 3 1x x x+ + + 21. ________________

22. ( ) ( )27 4 5 4 5p p p! ! ! 22. ________________


170

Factor by grouping. 23. 3 22 5 10x x x+ + + 23. ________________

24. 3 22 2 3 3z z z+ + + 24. ________________

25. 3 28 20 10 25p p p! + ! 25. ________________

26. 3 220 15 8 6x x x! + ! 26. ________________

27. 3 23 6 2x x x! ! + 27. ________________

28. 3 23 7 21x x x+ ! ! 28. ________________



171

Chapter 5 POLYNOMIALS: FACTORING 5.2 Factoring Trinomials of the Type 2x bx c+ + Learning Objective a Factor trinomials of the type 2x bx c+ + by examining the constant term c . Key Terms Use the vocabulary terms listed below to complete each statement in Exercises 1–4.

leading coefficient negative positive prime 1. In the expression 2 ,ax 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 of

the binomial factors have the same sign. 4. When the constant term of a trinomial is ___________________, the constant terms of

the binomial factors have opposite signs. Objective a Factor trinomials of the type 2x bx c+ + by examining the constant term c . Factor. Remember that you can check by multiplying. 5. 2 9 18x x+ +

Pairs of Factors

Sums of Factors

5. ________________


172

6. 2 8 16x x! +

Pairs of Factors

Sums of Factors

6. ________________

7. 2 20a a+ ! Pairs of Factors

Sums of Factors

7. ________________

8. 2 9 8y y! + Pairs of Factors

Sums of Factors

8. ________________

9. 2 13 30t t! + 9. ________________

10. 2 5 14p p! ! 10. ________________

11. 2 3 1x x+ + 11. ________________



173

12. 3 25 6x x x! + 12. ________________

13. 3 23 40x x x! ! 13. ________________

14. 216 60x x+ + 14. ________________

15. 4 27 10t t+ + 15. ________________

16. 215 5p p! + 16. ________________

17. 2 22 121x x+ + 17. ________________

18. 227 6a a! ! 18. ________________

19. 4 3 213 30x x x! ! 19. ________________


174

20. 2 16 60y y! + 20. ________________

21. 275 28t t! + 21. ________________

22. 2 0.1 0.06x x! ! 22. ________________

23. 260 7t t! ! 23. ________________

24. 2 220p pq q! ! 24. ________________

25. 2 25 66m mn n! ! 25. ________________

26. 8 7 65 15 140a a a! ! 26. ________________



175

Chapter 5 POLYNOMIALS: FACTORING 5.3 Factoring 2 ,ax bx c+ + 1a " : The FOIL Method Learning Objective a Factor trinomials of the type 2 ,ax bx c+ + 1a " , using the FOIL method. Key Terms Use the vocabulary terms listed below to complete each statement in Exercises 1–4.

First Inside Last Outside 1. In the product ( ) ( )3 2 1 ,x x! + x and 2x are the ___________________ terms.

2. In the product ( ) ( )3 2 1 ,x x! + 3! and 1 are the ___________________ terms.

3. In the product ( ) ( )3 2 1 ,x x! + 3! and 2x are the ___________________ terms.

4. In the product ( ) ( )3 2 1 ,x x! + x and 1 are the ___________________ terms.

Objective a Factor trinomials of the type 2 ,ax bx c+ + 1a " , using the FOIL method. Factor. 5. 22 5 12x x! ! 5. ________________

6. 25 32 12x x! + 6. ________________

7. 26 17 5x x! + 7. ________________


176

8. 23 2 1x x! ! 8. ________________

9. 216 8 15x x+ ! 9. ________________

10. 218 15 2x x+ + 10. ________________

11. 221 58 21x x! + 11. ________________

12. 225 40 16x x! + 12. ________________

13. 27 5 2x x! ! 13. ________________

14. 28 28 36x x! ! 14. ________________



177

15. 215 55 50x x+ + 15. ________________

16. 21 3 2x x! + ! 16. ________________

17. 216 72 40x x! ! 17. ________________

18. 221 6 9t t! ! 18. ________________

19. 4 3 210 21 10p p p! ! 19. ________________

20. 4 214 23 3a a+ + 20. ________________


178

21. 281 36 4y y+ + 21. ________________

22. 216 21 25x x+ + 22. ________________

23. 2 26 11 10m mn n! ! 23. ________________

24. 2 220 29 6a ab b+ + 24. ________________

25. 2 214 23 3p pq q+ + 25. ________________

26. 2 216 8 8x xy y! ! 26. ________________



179

Chapter 5 POLYNOMIALS: FACTORING 5.4 Factoring 2 ,ax bx c+ + 1a " : The ac-Method Learning Objective a Factor trinomials of the type 2 ,ax bx c+ + 1a " , 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

multiplying split sum 1. 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 a Factor trinomials of the type 2 ,ax bx c+ + 1a " , using the ac-method. Factor. Note that the middle term has already been split. 7. 2 4 3 12x x x+ + + 7. ________________

8. 26 4 15 10x x x+ + + 8. ________________


180

9. 224 20 6 5y y y! + ! 9. ________________

10. 4 2 22 4 7 14x x x! ! + 10. ________________

Factor using the ac-method. 11. 23 13 10x x+ ! 11. ________________

12. 28 26 15x x+ + 12. ________________

13. 24 8 21x x+ ! 13. ________________

14. 24 16 9x x! ! 14. ________________

15. 214 65 9x x! + 15. ________________

16. 225 30 9a a! + 16. ________________



181

17. 23 5 2a a! ! 17. ________________

18. 215 99 42x x+ ! 18. ________________

19. 215 22 9t t! ! 19. ________________

20. 24 7 2x x! ! 20. ________________

21. 3 26 19 15t t t+ + 21. ________________

22. 226 6 8y y! ! 22. ________________

23. 4 3 212 4 5p p p! ! 23. ________________

24. 3 2105 50 5x x x! + 24. ________________


182

25. 4 28 26 15x x! + 25. ________________

26. 216 72 81x x+ + 26. ________________

27. 215 10 18y y+ + 27. ________________

28. 2 215 11 2a ab b+ + 28. ________________

29. 2 26 20 14s st t+ + 29. ________________

30. 2 28 28 24x xy y! + 30. ________________



183

Chapter 5 POLYNOMIALS: FACTORING 5.5 Factoring Trinomial Squares and Differences of Squares Learning Objectives a Recognize trinomial squares. b Factor trinomial squares. c Recognize differences of squares. d Factor differences of squares, being careful to factor completely. Key Terms Use the vocabulary terms listed below to complete each statement in Exercises 1–4.

difference of squares

factored completely

sum of squares trinomial square 1. The expression 2 10 25x x+ + is a(n) ___________________. 2. The expression 29 16x ! is a(n) ___________________. 3. The expression 2 81y + is a(n) ___________________. 4. When no factor can be factored further, we have ___________________. Objective a Recognize trinomial squares. Determine whether each of the following is a trinomial square. 5. 2 10 25x x+ ! 5. ________________

6. 24 12 9x x! + 6. ________________


184

Objective b Factor trinomial squares. Factor completely. Remember to look first for a common factor and to check by multiplying. 7. 2 6 9x x! + 7. ________________

8. 236 12x x+ + 8. ________________

9. 4 210 25m m+ + 9. ________________

10. 212 84 147a a! + 10. ________________

11. 3 220 100x x x! + 11. ________________

12. 264 80 25x x! + 12. ________________

13. 2 24 20 25m mn n+ + 13. ________________

14. 4 24 16 16p p+ + 14. ________________

15. 2 28 16x xy y! + 15. ________________

16. 2 281 18a ab b! + 16. ________________



185

Objective c Recognize differences of squares. Determine whether each of the following is a difference of squares. 17. 2 100x ! 17. ________________

18. 2 24 10x y! + 18. ________________

19. 2 16x + 17. ________________

Objective d Factor differences of squares, being careful to factor completely. Factor completely. Remember to look first for a common factor. 20. 2 81a ! 20. ________________

21. 2100 y! + 21. ________________

22. 2 264x y! 22. ________________


186

23. 2 24 49m n! 23. ________________

24. 218 32x ! 24. ________________

25. 20.04 0.0025a ! 25. ________________

26. 436 1p ! 26. ________________

27. 43 48x ! 27. ________________

28. 16 81x ! 28. ________________

29. 219

16x!

29. ________________



187

Chapter 5 POLYNOMIALS: FACTORING 5.6 Factoring: A General Strategy Learning Objective a Factor polynomials completely using any of the methods considered in this chapter. Key Terms Use the vocabulary terms listed below to complete each statement in Exercises 1–4.

completely difference grouping square 1. When factoring a polynomial with two terms, determine whether you have

a(n) ___________________ of squares.

2. When factoring a polynomial with three terms, determine whether the trinomial is

a(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 in this chapter. Factor completely. 5. 25 20t ! 5. ________________

6. 2 18 81x x! + 6. ________________


188

7. 26 11 10x x! ! 7. ________________

8. 3 23 3x x x! ! + 8. ________________

9. 4 2 35 4 20x x x x+ ! ! 9. ________________

10. 3 26 4 10x x x+ ! 10. ________________

11. 2 1n + 11. ________________

12. 224 6a ! 12. ________________

13. 4 3 22 12 18x x x+ + 13. ________________

14. 2 3 7x x+ + 14. ________________

15. 225 45 10x x! ! 15. ________________



189

16. 53 3x x! 16. ________________

17. 3 33 12p q pq+ 17. ________________

18. ( ) ( )2 24 3 3x u v u v+ ! + 18. ________________

19. 2a a ab b! + ! 19. ________________

20. 3 5 2 3 4 420 5 15m n m n m n+ ! 20. ________________

21. 2 29 4 12c d cd+ ! 21. ________________

22. 2 2 225 40 16x z xyz y+ + 22. ________________

23. 2 24 60m mn n! ! 23. ________________


190

24. 2 2 2 15a b ab+ ! 24. ________________

25. 5 2 4 34 32x y x y x+ ! 25. ________________

26. 2 214

r t! 26. ________________

27. 8 4 81c d ! 27. ________________

28. 4 3 2 249 70 25m m n m n! + 28. ________________



191

Chapter 5 POLYNOMIALS: FACTORING 5.7 Solving Quadratic Equations by Factoring Learning Objectives a Solve equations (already factored) using the principle of zero products. b Solve equations by factoring and then using the principle of zero products. Key Terms Use the vocabulary terms listed below to complete each statement in Exercises 1–4.

factor products quadratic equation zero 1. 23 5 3 0x x! + = is an example of a(n) ___________________. 2. The principle of zero ___________________ states that a product is 0 if and only if one

or both of the factors is 0. 3. To use the principle of zero products, you must have ___________________ on one side

of the equation. 4. To use the principle of zero products, we ___________________ a quadratic

polynomial. Objective a Solve equations (already factored) using the principle of zero products. Solve using the principle of zero products. 5. ( )( )3 10 0x x+ ! = 5. ________________

6. ( )( )5 7 0x x+ + = 6. ________________


192

7. ( )15 0x x + = 7. ________________

8. ( )0 8x x= ! 8. ________________

9. ( ) ( )4 3 2 10 0x x+ ! = 9. ________________

10. ( ) ( )6 11 8 3 0x x! ! = 10. ________________

11. 1 1

3 4 02 3

x x§ ·§ ·! ! =¨ ¸¨ ¸© ¹© ¹

11. ________________

12. ( )( )0.1 0.2 0.5 15 0x x+ ! = 12. ________________



193

Objective b Solve equations by factoring and then using the principle of zero products. Solve by factoring and using the principle of zero products. Remember to check. 13. 2 5 4 0x x+ + = 13. ________________

14. 2 4 21 0x x+ ! = 14. ________________

15. 2 10 0x x! = 15. ________________

16. 2 12 0x x+ = 16. ________________

17. 2 25x = 17. ________________

18. 216 1 0x ! = 18. ________________


194

19. 20 4 4x x= + + 19. ________________

20. 23 10x x= 20. ________________

21. 23 5 2x x+ = 21. ________________

22. 220 11 3n n= + 22. ________________

23. 22 16 30 0y y+ + = 23. ________________

24. ( )3 5 28t t + = 24. ________________



195

Find the x-intercepts for the graph of the equation. (The grids are intentionally not included.) 25. 25. ________________

26. 26. ________________


196

27. Use the following graph to solve 2 2 0x x! ! = .

27. ________________

28. Use the following graph to solve 2 2 3 0x x! ! + = .

28. ________________



197

Chapter 5 POLYNOMIALS: FACTORING 5.8 Applications of Quadratic Equations Learning Objectives a Solve applied problems involving quadratic equations that can be solved by factoring. Key Terms Use the vocabulary terms listed below to complete each statement in Exercises 1–4.

hypotenuse legs Pythagorean right 1. 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, 2 2 2.a b c+ = Objective a Solve applied problems involving quadratic equations that can be solved by factoring. Solve. 5. The length of a rectangular table is 5 ft greater than the

width. The area of the table is 24 ft!. Find the length and the width.

5. ________________


198

6. A rectangular serving tray is twice as long as it is wide. The area of the tray is 338 in!. Find the dimensions of the tray.

6. ________________

7. A triangle is 8 cm wider than it is tall. The area is 42 cm!. 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 m!. Find the base and the height.

8. ________________

9. A soccer league has 12 teams. What is the total number of games to be played if each team plays every other team twice? Use 2 .x x N! =

9. ________________

10. A softball league plays a total of 210 games. How many teams are in the league if each team plays every other team twice? Use 2 ,x x N! = where N is the number of games and x is the number of teams.

10. ________________



199

11. There are 30 people at a party. How many handshakes are

possible? Use ( )21,

2N x x= ! where N is the number of

handshakes and x is the number of people.

11. ________________

12. Everyone at a meeting shook hands with each other. There were 276 handshakes in all. How many people were at the

meeting? Use ( )21,

2N x x= ! where N is the number of

handshakes and x is the number of people.

12. ________________

13. During a toast at a party, there were 66 “clinks” of glasses. How many people took part in the toast? Use

( )21,

2N x x= ! where N is the number of clinks and x is

the number of people.

13. ________________

14. The product of the page numbers on two facing pages of a book is 110. Find the page numbers.

14. ________________

15. The product of two consecutive even integers is 288. Find the integers.

15. ________________

16. The product of two consecutive odd integers is 195. Find the integers.

16. ________________


200

17. The length of one leg of a right triangle is 15 ft. The length of the hypotenuse is 5 ft longer than the other leg. Find the length of the hypotenuse and the other leg.

17. ________________

18. A metal sign is a right triangle that is 30 in. tall and has a hypotenuse of length 34 in. If the metal costs $0.10 per square inch, find the cost of the sign.

18. ________________

19. Johann landed his Cessna 172, traveling 13,000 ft over a 12,000-ft horizontal distance. From what altitude did the descent begin?

19. ________________

20. The guy wire on a tower is 10 ft longer than the height of the tower. If the guy wire is anchored 50 ft from the foot of the tower, how tall is the tower?

20. ________________

21. A model rocket is launched with an initial velocity of 160 ft/sec. Its height ,h in feet, after t seconds is given by the

formula 2160 16 .h t t= ! After how many seconds will the rocket first reach a height of 336 ft?

21. ________________

22. The sum of the squares of two consecutive odd positive integers is 202. Find the integers

22. ________________


chapter 5 polynomials: factoring - ms. wilson's...

Documents