College Algebra 2Nd Abramson Solutions Manual by mmsharp

Solutions

Manual for College Algebra 2nd Edition by Abramson, OpenTax

ISBN: 9781998109425

Chapter 1 Prerequisites

1.1 Real Numbers: Algebra Essentials

Section Exercises

Verbal

1. Is 2 an example of a rational terminating, rational repeating, or irrational number? Tell why it fits that category.

Irrational number. The square root of two does not terminate, and it does not repeat a pattern. It can’t be written as a quotient of two integers, so it is irrational.

2. What is the order of operations? What acronym is used to describe the order of operations, and what does it stand for?

The order of operations is a set of rules that prioritize the different operations such that expressions can be evaluated consistently. PEMDAS stands for Parentheses, Exponents, Multiplication and Division, Addition and Subtraction.

3. What do the Associative Properties allow us to do when following the order of operations? Explain your answer.

The Associative Properties state that the sum or product of multiple numbers can be grouped differently without affecting the result. This is because the same operation is performed (either addition or subtraction), so the terms can be re-ordered.

Numeric

For the following exercises, simplify the given expression.

6. ( )3 1868 +− 318(2) 18(8) 188 10 +− +−

9. ( )358 3(3) 9

10. 46102 +− 465 105 5 +−

11. ( ) 123696 + 12(4)6 36 9 + +

)

13. 312219 −+ 32419 2119 2 −+ −+

14. 2874 + 2564 214 16 + +

15. ( ) 56411 ++− 5(10)11 1511 4 +−

16. 29183 − 9189 92 7 −

17. 14376 − 4276 66 0 −

18. ( ) 93112 −+ 9(14)2 928 19 −

19. 6221 +− 641 101 9 +−

20. ( )64842 +

64(88) 64(16) 4 + 

21. ( ) 2942 + 94(4) 916 25 + +

22. ( )2 1233  ( )2 2 43 12 144 

23. 2 2557 − 25257 17 6 −

24. ( ) ( )15737 −− 8(37) 84 32 − −

25. ( )2491 −− 89(1) 89 17 + 26. 2 1 425 5 − 1 1625 5 165 11 −

27. ( ) 12316 − 12(2)6 246 4 

Algebraic

For the following exercises, evaluate the expression for the given value of the variable.

28. 8(3)64 x +− for 2 x = 24

29. 482yy +− for 3 y = 14

30. (113)184 aa+−+ for 2 a =− 21

31. 42(14)36 zz−+− for 5 z = 66

32. 2 4(72)200 y −+ for 2 y =− 0

33. 2 (2)13 x −++ for 2 x = 12

34. For the 8(24)15bb+−+ for 3 b =− 90

35. 2(114)36 c for 0 c = 44

36. 4(31)4 x for 10 x = 76

37. ( ) 2 1 84 4 w for 1 w = 2

For the following exercises, simplify the expression.

38. ( )4137 xx+−

4x + x 13- 7 ( ) Combine terms inside of parentheses

4x + x(6) Rewrite

4x + 6x Combine like terms

10x

39. ( )2 2411 yy

( )2 2411 Squarethe4

21611 Combineliketerms 1411 yy yy y

40. ( ) 3 64126 2 a a − ( ) 3 64126 Cube the 2 2

82 Combine like terms 6 a a

(64)126 Multiply64 by1/8

8 (8)126 Rewrite and divide 12 by6

41. ( ) 8431 bb−+ ( ) 8431 Multiplythecoefficients ofb

8121 Combineliketerms 41 bb bb b −+ −+ −+

42. ( )5396 ll−

5l ¸ 3l ´ 9 - 6 ( ) Combine terms inside of parentheses

5l ¸ 3l ´ 3 Divide 5l by 3l 5 3 ´ 3 Multiply 5/3 by 3 5

43. 2736 zz−+ 2736 Squarethe6

7336 Multiply36andz

7336 Combineliketerms 433 zz zz zz z −+ −+ −+

44. 4318912 x +− 4318912 Multiply4and3 1218912 Divide18xby9

Combineliketerms

45.

)

47. 61236bb+− 61236 Multiply3 and 6b 61218 Combine like terms

48.

)

50. ( ) ( )8318 m −+−

( ) ( )8318 Distributethe8

2481(8) Multiplythe8and-1

2488 Combineliketermsandrewrite

816 m m m m −+− −+− −+

51. ( ) ( ) 9423423 xxxx ++−+

9x + 4x 2 + 3 ( ) - 4 2x + 3x ( ) Combine terms inside of the parentheses

9x + 4x(5) - 4(5x) Distribute the 4x and the -4

9x + 20x - 20x Combine like terms

52. ( ) 2 543x ( ) 2 543 Square the 5 and multiply4 by3x

1225 x x x −+

2512 Rewrite

Real-World Applications

For the following exercises, consider this scenario: Fred earns $40 at the community garden. He spends $10 on a streaming subscription, puts half of what is left in a savings account, and gets another $5 for walking his neighbor’s dog.

53. Write the expression that represents the number of dollars Fred keeps (and does not put in his savings account). Remember the order of operations. ( )

Note that Fred earns 40 then spends 10 before putting half inside of his bank account. The additional 5 comes after the deposit, so we write the expression: 1 40105 2 −+

54. How much money does Fred keep? 1 (4010)5 Parenthesisfirst 2 1 (30)5 Multiplication 2 155 Additionlast 20 −+ + +

55. According to the U.S. Mint, the diameter of a quarter is 0.955 inches. The circumference of the quarter would be the diameter multiplied by . Is the circumference of a quarter a

whole number, a rational number, or an irrational number?

is irrational and 0.955 is rational, so their product is an irrational number.

56. Jessica and her roommate, Adriana, have decided to share a change jar for joint expenses. Jessica put her loose change in the jar first, and then Adriana put her change in the jar. We know that it does not matter in which order the change was added to the jar. What property of addition describes this fact? The commutative property of addition

For the following exercises, consider this scenario: There is a mound of g pounds of gravel in a quarry. Throughout the day, 400 pounds of gravel is added to the mound. Two orders of 600 pounds are sold and the gravel is removed from the mound. At the end of the day, the mound has 1,200 pounds of gravel.

57. Write the equation that describes the situation.

( ) 40026001200 g +−=

58. Solve for g.

( ) 40026001200 Distributethe2

2000 g g g

40012001200 Add 1200 tobothsides

4002400 Subtract 400frombothsides

59. Ramon runs the marketing department at their company. Their department gets a budget every year, and every year, they must spend the entire budget without going over. If Ramon spends less than the budget, then their department gets a smaller budget the following year. At the beginning of this year, Ramon got $2.5 million for the annual marketing budget. They must spend the budget such that 2,500,0000 x −= . What property of addition tells us what the value of x must be? Inverse property of addition

Technology

For the following exercises, use a graphing calculator to solve for x. Round the answers to the nearest hundredth.

60. ( )2 3 0.512.348 5 x −= 1.56

61. ( )2 0.250.757.29.9 x −−= 68.4

Extensions

62. If a whole number is not a natural number, what must be the number?

0. Recall that the whole numbers are the natural numbers plus zero.

63. Determine whether the statement is true or false: The multiplicative inverse of a rational number is also rational.

True

64. Determine whether the statement is true or false: The product of a rational and irrational number is always irrational. False. 0 is rational and 0 times an irrational number is 0.

65. Determine whether the simplified expression is rational or irrational: ( )( ) 18451.

( )( )18451 Multiply4 and 5

1820(1) Multiply20 and -1

1820 Add -18 and 20 2 −+ Irrational

66. Determine whether the simplified expression is rational or irrational: ( ) 16455.−++ ( ) 16455 Multiply4and5

1625 Add-16and25 9 Evaluate 3 −++

16205 Add20and5

67. The division of two natural numbers will always be what type of number? Rational

68. What property of real numbers would simplify the following expression: ( )471 x +− ? Distributive property

Chapter 1 Prerequisites

1.2 Exponents and Scientific Notation

Section Exercises

Verbal

1. Is 32 the same as 23 ? Explain. No, the two expressions are not the same. An exponent tells how many times you multiply the base. So 32 is the same as 222  , which is 8. 23 is the same as 33  , which is 9.

2. When can you add two exponents? You can add two exponents when the bases are the same and the expressions are being multiplied.

3. What is the purpose of scientific notation? It is a method of writing very small and very large numbers.

4. Explain what a negative exponent does. A negative exponent switches a numerator to a denominator or a denominator to a numerator. It is the number of times you multiply 1 over the base.

Numeric

For the following exercises, simplify the given expression. Write answers with positive exponents.

For the following exercises, write each expression with a single base. Do not simplify further. Write answers with positive exponents.

For the following exercises, express the decimal in scientific notation.

21. 0.0000314 5 3.1410 

22. 148,000,000

1.4810 

For the following exercises, convert each number in scientific notation to standard notation.

23. 101.610  16,000,000,000

24. 9 9.810  0.0000000098

Algebraic

For the following exercises, simplify the given expression. Write answers with positive exponents. 25.

27.

33.

34. ( )2 lw  22lw

35. ( )3 714 yx  2114 21 14 yx y x  36. 2 32 a

37.

(25)(5)

45. A dime is the thinnest coin in U.S. currency. A dime’s thickness measures 3 1.3510  meters. Write the number in standard notation.

0.00135 meter

46. The distance between Earth and the Sun (on average) is 92,960,000 miles. Rewrite the distance using scientific notation.

79.296010  miles

47. A terabyte is made of approximately 1,099,500,000,000 bytes. Rewrite in scientific notation.

121.099510 

48. The Gross Domestic Product for the United States in the first quarter of 2014 was 13 $1.7149610.  Rewrite GDP in standard notation.

$17,149,600,000,000

49. One picometer is approximately 11 3.39710  inches. Rewrite the number of inches in a picometer using standard notation. 0.00000000003397 inches

50. The value of the services sector of the U.S. economy in the first quarter of 2012 was $10,633.6 billion. Rewrite this amount in scientific notation.

$ 131.0633610 

Technology

For the following exercises, use a graphing calculator to simplify. Round the answers to the nearest hundredth.

51. 2 333 3 12 4 m

12,230,590,464 66 m

52. 3231715 x  3 4913 225x

Extensions

For the following exercises, simplify the given expression. Write answers with positive exponents.

53.

58. Avagadro’s constant is used to calculate the number of particles in a mole. A mole is a basic unit in chemistry to measure the amount of substance. The constant is 236.022141310  . Write Avagadro’s constant in standard notation. 602,214,130,000,000,000,000,000

59. Planck’s constant is another important unit of measure in quantum physics. It describes the relationship between energy and frequency. The constant is written as 34 6.6260695710  . Write Planck’s constant in standard notation.

0.00000000000000000000000000000000062606957

Chapter 1

Prerequisites

1.3 Radicals and Rational Expressions

Section Exercises

Verbal

1. What does it mean when a radical does not have an index? Is the expression equal to the radicand? Explain. When there is no index, it is assumed to be 2 or the square root. The expression would only be equal to the radicand if the index were 1.

2. Where would radicals come in the order of operations? Explain why. Radicals are another way of writing fractional exponents, so they would come after performing operations within parenthesis but before multiplication and division.

3. Every number will have two square roots. What is the principal square root? The principal square root is the nonnegative root of the number.

4. Can a radical with a negative radicand have a real square root? Why or why not? A radical with a negative radicand cannot have a real square root. This is because the roots must be congruent, but the product of two negative numbers is positive, so we cannot multiply two real numbers that are the same and get a negative product.

Numeric

For the following exercises, simplify each expression. 5.

7. ( )4916 + ( )4916 Combinetermsinsideofparenthesis 4(25) Separateintotwodifferent squareroots

Evaluatesqaureroots

Multiply

Notethat289 is17 and 121is11

Multiplicationpropertyofradicals

49isasquareroot; 2isnot aperfect square

146624 24 has factors 4 and 6

146646 Multiplication propertyof radicals

146646 4 is a perfect square; 6 is not a perfect square

1466(2)6 Multiply 146126 Subtract

20. 155745 +

155745 45 has factors 9 and 5

155795 Multiplication propertyof radicals

155795 9 is a perfect square; 5 is not a perfect square

1557(3)5 Multiply 155215 Add

23. ( )( )4230 ( )( ) ( )( ) 4230 42hasfactors7and6; 30has factors5and6 7656 Multiplicationpropertyofradicals 7566 Multiplicationpropertyofradicals 7566 Evaluateandrewrite

24. 123475

123475 75 has factors 25 and 3

1234253 Multiplication propertyof radicals

1234253 25 is a perfect square; 3 is not a perfect square

1234(5)3 Multiply 123203 Subtract

34. 33 343216 −+

3333 343216 432 has factors 216 and 2; 16 has factors 8 and 2 3216282 Multiplication propertyof radicals 3216282 216 is a perfect

Algebraic

For the following exercises, simplify each expression.

35.

400Multiplication property of radicals

400400 is a perfect square; xis a perfect square 20 x x x

36. 2 4y 2 22 4Multiplication property of radicals

44 is a perfect square; yis a perfect square 2 y y y

37. 49 p

49Multiplication property of radicals

4949 is a perfect square; p is not a perfect square 7 p p p

38. ( ) 1 26 2 144 pq ( ) 1 26 2 26 3 144 Rewritewith squareroots 144 144, p, and q areperfect squares 12 pq pq pq

39. 5 2 289 m 55 22 414 2 222 2 289 Usepropertiesofexponentsto rewritem 289 m; 289isaperfect square 17 m mmm mm =

40. 2 9327 m + 2 22 9327

Multiplication propertyof radicals; 27 has factors 9 and 3

9393 m is a perfect square; multiplication propertyof radicals 9393 9 is a perfect square; 3 is not a perf m m m + + + ect square 9333 m +

41. 2 3 abba 2 22 3 Multiplication propertyof addition 3 b is a perfect square 3 Subtract 2 abba abba baba

Rewrite q; 63p has factors 9 and 7p

Simplify; Multiplication propertyof radicals

9 is a perfect square; q and 7p are not perfect squares

Multiplytopand bottombytheconjugateofthedenominator

62.

Real-World Applications

65. A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So the length of the guy wire can be found by evaluating

90,000160,000 + . What is the length of the guy wire?

90,000160,000 Add

Note:250,000=500 500feet +

250,000

66. A car accelerates at a rate of

6m/s t where t is the time in seconds after the car moves from rest. Simplify the expression.

Extensions

For the following exercises, simplify each expression. 67.

2 Multiplythetopandbottombytheconjugateofthedenom.

Give 2 acommondenominator

68.

Rewriteallfractionalexponents

4and16areperfect squares; 8isaperfect cube

Evaluateexponentialexpressions

Multiplicationpropertyofradicals; multiply

Evaluatesquarerootsandcombineliketerms

Multiplytopandbottombytheconjugateofthedenominator

Multiply; UsetheFOILmethod

Combineliketerms

Chapter 1 Prerequisites

1.4 Polynomials

Section Exercises

Verbal

1. Evaluate the following statement: the degree of a polynomial in standard form is the exponent of the leading term. Explain why the statement is true or false.

The statement is true. In standard form, the polynomial with the highest value exponent is placed first and is the leading term. The degree of a polynomial is the value of the highest exponent, which in standard form is also the exponent of the leading term.

2. Many times, multiplying two binomials with two variables results in a trinomial. This is not the case when there is a difference of two squares. Explain why the product in this case is also a binomial.

In a difference of two squares, the products are additive inverses of each other so when they are added, their sum is zero.

3. You can multiply polynomials with any number of terms and any number of variables using four basic steps over and over until you reach the expanded polynomial. What are the four steps?

Use the distributive property, multiply, combine like terms, and simplify.

4. State whether the following statement is true and explain why or why not: a trinomial is always a higher degree than a monomial.

The statement is false. The degree doesn’t have anything to do with the number of terms in a polynomial; it is the highest power of the variable that occurs in the polynomial. So a monomial can have a power of 4 while a trinomial may have a power of 2.

Algebraic

For the following exercises, identify the degree of the polynomial.

9. 2 44xx++ 2

10. 45 634 yyy−+− 5

For the following exercises, find the sum or difference.

11. ( ) ( ) 22 123819 xxx+−−

( ) ( ) 22 22 2 123819 Distributetheminussymbol 123819 Combineliketerms 4319 xxx xxx xx +−− +−+ ++

12. ( ) ( ) 322 4826 zzzzz +−+−++

( ) ( ) 322 322 32 4826 Rewrite 4826 Combineliketerms 466 zzzzz zzzzz zz +−+−++ +−−++ ++

13. ( ) ( ) 22 62424363 wwww ++−−+

( ) ( ) 22 22 2 62424363 Distributetheminussymbol 62424363 Combineliketerms 33021 wwww wwww ww ++−−+ ++−+− ++

14. ( ) ( ) 3232 7641334617 aaaaaa +−−+−−++

( ) ( ) 3232 3232 33 7641334617 Rewrite 7641334617 Combineliketerms 4224 aaaaaa aaaaaa aaa +−−+−−++ +−−−−++ +++

15. ( ) ( ) 43232 1161848363 bbbbbbb −+−+−++ ( ) ( ) 43232 43232 432 1161848363 Distributetheminussymbol 1161848363 Combineliketerms 1191278 bbbbbbb bbbbbbb bbbb −+−+−++ −+−+−−− −+−+

16. ( ) ( ) 242 4925163216 ppp−+−+

( ) ( ) 242 242 42 4925163216Rewrite 4925163216Combine like terms and rewrite 16179 ppp ppp pp −+−+ −+−+ +−

For the following exercises, find the product.

17. ( )( )4264 xx+−

( )( ) 2 2 4264

2416128 Combineliketerms 2448 xx xxx xx +− −+−

18. ( )( ) 22 14423 cccc +−

UsetheFOILmethod

( )( ) 22 4332 432 14423

UsetheFOILmethod 1842812 Combineliketerms 183412 cccc cccc ccc +− −+−

19. ( )( ) 22 6644 bb

( )( ) 22 422 42 6644

244824 bb bbb bb −−+ −+

UsetheFOILmethod 24242424 Combineliketerms

20. ( )( )3529 dd−+

( )( ) 2 2 3529

61745 dd ddd dd −+ +−− +−

UsetheFOILmethod

6271045 Combineliketerms

21. ( )( )911119 vv

( )( ) 2 2 911119 UsetheFOILmethod 998112199 Combineliketerms 9920299 vv vvv vv −−+ −+

22. ( )( ) 22 4734 ttt+−+

( )( ) 22 423 432 4734 UsetheFOILmethod 12162128 Rewrite 12211628 ttt tttt tttt +−+ −+−+ −−++

23. ( )( ) 2 849 nn−+

( )( ) 2 32 32 849 UsetheFOILmethod

872436 Rewrite 847236 nn nnn nnn −+ +−− −+−

For the following exercises, expand the binomial.

24. ( )2 45 x +

( )2 2 2 45

Expandtheexpression (45)(45) UsetheFOILmethod

16202025 Combineliketerms

164025 x xx xxx xx + ++ +++ ++

25. ( )2 37 y

( )2 2 2 37

9212149 Combineliketerms 94249 y yy yyy yy −−+ −+

Expandtheexpression (37)(37) UsetheFOILmethod

26. ( )2 124x

( )2 2 2 124

Expand theexpression (124)(124) UsetheFOILmethod

144484816 Combineliketerms and rewrite 1696144 x xx xxx xx −−+ −+

27. ( )2 49 p +

( )2 2 2 49

Expandtheexpression (49)(49) UsetheFOILmethod

16363681 Combineliketerms

167281 p pp ppp pp + ++ +++ ++

28. ( )2 23 m

( )2 2 2 23

Expandtheexpression (23)(23) UsetheFOILmethod

4669 Combineliketerms

4129 m mm mmm mm −−+ −+

29. ( )2 36 y

( )2 2 2 36

Expandtheexpression (36)(36) UsetheFOILmethod

9181836 Combineliketerms

93636 y yy yyy yy −−+ −+

30. ( )2 91 b +

81991 Combineliketerms

( )2 2 2 91 Expand theexpression (91)(91) UsetheFOILmethod

81181 b bb bbb bb + ++ +++ ++

For the following exercises, multiply the binomials.

31. ( )( )4141 cc+−

( )( ) 2 2 4141 Use the FOILmethod

16441 Combine like terms 161 cc ccc c +− −+−

32. ( )( )9494 aa−+ ( )( ) 2 2 9494 UsetheFOILmethod

81363616 Combineliketerms

8116 aa aaa a −+ +−−

33. ( )( )156156 nn−+

225909036 Combine like terms

( )( ) 2 2 156156 Use the FOILmethod

22536 nn nnn n −+ +−−

34. ( )( )252252 bb+−

( )( ) 2 2 252252

6254 bb bbb b +− −+−

UsetheFOILmethod

62550504 Combineliketerms

35. ( )( ) 4444mm+−

( )( ) 2 2 4444

UsetheFOILmethod 16161616 Combineliketerms 1616 mm mmm m +− −+− −+

36. ( )( )147147 pp+−

( )( ) 2 2 147147

196989849 Combineliketerms 19649 pp ppp p +− −+−

UsetheFOILmethod

37. ( )( )11101110 qq−+

( )( ) 2 2 11101110

121100 qq qqq q −+ +−−

UsetheFOILmethod 121110110100 Combineliketerms

For the following exercises, multiply the polynomials.

38. ( )( ) 2 22141 xxx++−

( )( ) 2 322 322 22141

Usethedistributiveproperty 884221 Combineliketerms 8621 xxx xxxxx xxx ++− ++−−− ++−

39. ( )( ) 22 4741 ttt+−− ( )( ) 22 4322 432 4741

Usethedistributiveproperty 1642847 Combineliketerms 164327 ttt ttttt tttt +−− +−−−+ +−−+

40. ( )( ) 2 121xxx−−+

( )( ) 2 322 32 121

Usethedistributiveproperty 221 Combineliketerms 331 xxx xxxxx xxx −−+ −+−+− −+−

41. ( )( ) 2 249yyy

( )( ) 2 322 32

492818Combine like terms 618 yyy yyyyy yyy −−−++ −−+

249Use the distributive property

42. ( )( ) 2 65651 kkk−+−

( )( ) 2 322 3 65651

36315 kkk kkkkk kk −+− +−−−+ −+

Usethedistributiveproperty

3630630255 Combineliketerms

43. ( )( ) 2 32101 ppp+−−

( )( ) 2 322 32 32101Use the distributive property

31210 ppp ppppp ppp +−− −+−−+ −−+

33221010Combine like terms

44. ( )( ) 2 413279 mmm−−+

( )( ) 2 322 32 413279 Usethedistributiveproperty 828362691117 Combineliketerms

854127117 mmm mmmmm mmm −−+ −+−+− −+−

45. ( )( )abab +−

( )( ) 22 22

UsetheFOILmethod Combineliketerms abab aababb ab +− −+−

46. ( )( ) 4664 xyxy

( )( ) 22 22 4664

245224 xyxy xxyxyy xxyy −−+ −+

UsetheFOILmethod

24163624 Combineliketerms

47. ( )2 45tu ( )2 22 22 45

Expand theexpression (45)(45) UsetheFOILmethod

16202025 Combineliketerms

164025 tu tutu ttutuu ttuu −−+ −+

48. ( )( )94128 mnm+−+

( )( ) 2 2 94128

187283228 Combineliketerms 18708328 mnm mmmnnm mmmnn +−+ +++−− +++−

Usethedistributiveproperty

49. ( )( )41 txtx−−+

( )( ) 22 22 41

Usethedistributiveproperty 444 Combineliketerms 445 txtx ttxttxxx txttxx −−+ −+−+− ++−−

50. ( )( ) 222 12 baabb −++ ( )( ) 222 223422 432222 12

Usethedistributiveproperty 22 Rewrite 22 baabb ababbaabb bababbaba −++ ++−−− ++−−−

51. ( )( ) 467 rdrd −+ ( )( ) 22 22 467

UsetheFOILmethod 242867 Combineliketerms 24227 rdrd rrdrdd rrdd −+ +−− +−

52. ( )( ) 22 xyxxyy +−+ ( )( ) 22 322223 33

Use the distributive property Combine like terms xyxxyy xxyxyxyxyy xy +−+ −++−+ +

Real-World Applications

53. A developer wants to purchase a plot of land to build a house. The area of the plot can be described by the following expression: ( )( )4183 xx+− where x is measured in meters.

Multiply the binomials to find the area of the plot in standard form.

( )( ) 2 2 4183

3243squaremeters xx xxx xx +− −+−

UsetheFOILmethod

321283 Combineliketerms

54. A prospective buyer wants to know how much grain a specific silo can hold. The area of the floor of the silo is( )2 29 x + . The height of the silo is 1010 x + , where x is measured in feet. Expand the square and multiply by the height to find the expression that shows how much grain the silo can hold.

2x + 9 ( )2

(2x + 9)(2x + 9)

4x 2 +18x +18x + 81

4x 2 + 36x + 81(10x +10)

40x 3 + 40x 2 + 360x 2 + 360x + 810x + 810

40x 3 + 400x 2 +1170x +810 cubic feet

Extensions

Expand the expression

Use the FOIL method

Combine like terms

Multiply by the height using FOIL

Combine like terms

For the following exercises, perform the given operations.

55. ( ) ( ) ( ) 2 2 47214211 tttt −+−++

4t - 7 ( )2 2t +1 ( ) - 4t2 + 2t +11 ( )

(4t - 7)(4t - 7) 2t +1 ( ) - 4t2 + 2t +11 ( )

16t2 - 28t - 28t + 49 2t +1 ( ) - 4t2 + 2t +11 ( )

(16t2 - 56t + 49) 2t +1 ( ) - 4t2 + 2t +11 ( )

32t3 +16t2 -112t2 - 56t + 98t + 49 - 4t2 + 2t +11 ( )

32t3 - 96t2 + 42t + 49 - 4t2 - 2t -11

32t3 -100t2 + 40t + 38

Expand the first expression

Use the FOIL method

Combine like terms

Use the distributive property

Combine like terms

56.

UsetheFOILmethod

Chapter 1 Prerequisites

1.5 Factoring Polynomials

Section Exercises

Verbal

1. If the terms of a polynomial do not have a greatest common factor, then does that mean it is not factorable? Explain. The terms of a polynomial don’t have to have a common factor for the entire polynomial to be factorable. For example, 2 4x and 2 9y don’t have a common factor, but the whole polynomial is still factorable ( )( ) 22 492323 xyxyxy −=+− .

2. A polynomial is factorable, but it is not a perfect square trinomial or a difference of two squares. Can you factor the polynomial without finding the greatest common factor? Yes, you would just have to use extra steps to factor out multiple common terms.

3. How do you factor by grouping?

Divide the x term into the sum of two terms, factor each portion of the expression separately, and then factor out the GCF of the entire expression.

Algebraic

For the following exercises, find the greatest common factor.

4. 2 14418xxyxy +−

5. 222493577mbmbama −+

6. 32233045135xyxyxy −+ 15xy

7. 332332003040 pmpmm −+

3 10m

8. 423324361854jkjkjk −+ 22 18 jk

9. 432623yyyy −+− y For the following exercises, factor by grouping.

10. 2 654 xx+− ( ) ( ) ( )( ) 2 2 654

Rewrite 5 as 8 - 3 (68)(34) FactorouttheGCFfromeach expression 23434 Usethedistributiveproperty 2134 xxxxx xxx xxx xx +− ++−− +−+ −+

11. 2 2918 aa+− 2 2 2918

Rewrite 9 as12 - 3 (212)(318) Factorout theGCFfromeachexpression 2(6)3(6) Usethedistributive property (23)(6) aaaaa aaa aaa aa +− ++−− +−+ −+

12. 2 64163 cc++

2 2 64163

Rewrite 41 as 27 14 (614)(2763) Factorout theGCFfromeachexpression

2(37)9(37) Usethedistributiveproperty (29)(37) ccxxx ccc ccc cc +++ +++ +++ ++

13. 2 61911 nn

2 2 61911

Rewrite -19 as 3 - 22 (63)(2211) Factorout theGCFfromeachexpression

3(21)11(21) Usethedistributiveproperty (311)(21) nnnnn nnn nnn

14. 2 204724 ww−+

2 2 204724

Rewrite 47 as 1532 (2015)(3224) Factorout theGCFfrom eachexpression 5(43)8(43) Usethedistributiveproperty (58)(43) wwwww www www ww −+−−− −+−+

15. 2 257 pp

2 2 257

Rewrite 5 as 27 (22)(77) Factorout theGCFofeach expression 2(1)7(1) Usethedistributiveproperty (27)(1) ppppp ppp ppp pp ++−− +−+

For the following exercises, factor the polynomial.

16. 2 7487 xx+−

2 2 7487

Rewrite 48 as 49(749)(7) FactorouttheGCFofeach expression 7(7)(7) Usethedistributiveproperty (71)(7) xxxxx xxx xxx xx +− ++−− +−+

17. 2 1099 hh

2 2 1099

Rewrite 9 as-15 6 (1015)(69) Factorout theGCFofeach expression 5(23)3(23) Usethedistributiveproperty (53)(23) hhhhh

18. 2 225247 bb

2 2 225247

Rewrite -25 as-38 13 (238)(13247) Factorout theGCFofeach expression 2(19)13(19) Usethedistributiveproperty (213)(19) bbbbb

19. 2 9738 dd−+

2 2 9738

Rewrite -73 as -72(972)(8) FactorouttheGCFofeachexpression 9(8)(8) Usethedistributiveproperty (91)(8) ddddd ddd ddd dd

20. 2 9018190 vv−+

2 2 9018190

Rewrite -181 as -100 - 81 901008190 Factorout theGCFofeach expression 10(910)9(910) Usethedistributiveproperty (109)(910) vvvvv

21. 2 1213 tt+−

2 2 1213

Rewrite as13 - 12 (1213)(1213) Factorout theGCFofeach expression (1213)(1213) Usethedistributive property (1)(1213) ttttt

22. 2 215 nn

2 2 215

Rewrite - as -6 5 (26)(515) Factor out the GCFof each expression 2(3)5(3) Use thedistributiveproperty (25)(3) nnnnn nnn nnn nn −−+ −+− −+− +−

23. 2 16100 x

2 2 16100

Rewrite 0 as -40 40 (1640)(40100) Factorout theGCFofeachexpression 4(410)10(410) Usethedistributiveproperty (410)(410) xxxx

24. 2 25196 y 2 2 25196

Rewrite 0 as 70 - 70 (2570)(70196) Factorout theGCFofeachexpression 5(514)14(514) Usethedistributiveproperty (514)(514) yxxx yyy yyy yy ++−− +−+ −+

25. 2 121169 p 2 2 121169

Rewrite 0 as143 - 143 (121143)(143169) Factorout theGCFof eachexpression 11(1113)13(1113) Usethedistributiveproperty (1113)( pxpp ppp ppp p ++−− +−− + 1113) p

26. 2 49 m 2 2 49

Rewrite 0 as 6 - 6 (46)(69) FactorouttheGCFofeach expression 2(3)3(3) Usethedistributiveproperty (23)(3) mmmm mmm mmm mm ++−− +−+ −+

27. 2 36181 d

2 2 36181

Rewrite 0 as171 - 171 (361171)(17181) Factorout theGCFof eachexpression 19(199)9(199) Usethedistributiveproperty (199)(1 dddd ddd ddd d ++−− +−− 99) d +

28. 2 324121 x 2 2 324121

Rewrite 0 as198 - 198 (324198)(198121) Factorout theGCFofeach expression 18(1811)11(1811) Usethedistributiveproperty (18 xxxx xxx xxx x ++−− +−− 11)(1811) x +

29. 2214425bc 22 22 14425

Rewrite 0 as 60 - 60 (14460)(6025) FactorouttheGCFofeachexpression 12(125)5(125) Usethedistributiveproperty (12 bcbcbcbc bbcbcc bbccbc ++−− +−− 5)(125) bcbc −+

30. 2 1681 aa−+ 2 2 1681

Rewrite -8 as -4 - 4 (164)(41)

FactorouttheGCFofeachexpression 4(41)(41) Usethedistributiveproperty (41)(41) Rewrite (4 aaaaa aaa aaa aa a −+ −+−+ 21)

31. 2 49168144 nn++ 2 2 49168144 Rewrite168 as84 84 (4984)(84144) FactorouttheGCFofeachexpression 7(712)12(712) Usethedistributiveproperty (712)(712) nnnnn nnn nnn nn +++ +++ +++ ++ 2 Rewrite (712) n +

32. 2 1218816 xx−+

2 2 1218816

Rewrite88 as -44 - 44 (12144)(4416) FactorouttheGCFofeachexpression 11(114)4(114) Usethedistributiveproperty (114)(114) xxxxx

(114) x

33. 2 22512016 yy++

2 2 22512016

Rewrite120 as 60 60 (22560)(6016) FactorouttheGCFofeachexpression 15(154)4(154) Usethedistributiveproperty (154)(154) yyyyy

(154) y

34. 2 20100mm−+

2 2 20100

Rewrite -20 as -10 - 10 (10)(10100) FactorouttheGCFofeachexpression (10)10(10) Usethedistributiveproperty (10)(10) mmmmm

(10) m

35. 2 25120144 pp−+

2 2 25120144

Rewrite -120 as -60 - 60 (2560)(60144) FactorouttheGCFofeachexpression 5(512)12(512) Usethedistributiveproperty (512)(512) ppppp ppp ppp pp −+ −+−+

Rewrite (512) p

36. 2 366025 qq++ 2 2 366025 Rewrite 60 as30 30 (3630)(3025) FactorouttheGCFofeachexpression 6(65)5(65) Usethedistributiveproperty (65)(65) qqqqq qqq qqq qq +++ +++ +++ ++ 2 Rewrite (65) q +

For the following exercises, factor the polynomials.

37. 3 216 x +

(6)(636)

38. 3 278 y

and8=2 (32)(964) y yyy

39. 3 125343 a +

2 125343 Sumofcubes;125=5 and343=7 (57)(253549) a aaa + +−+

40. 33 8 bd

22 8 Differenceofcubes;8=2 (2)(24) bd bdbbdd −++

41. 3 64125 x

and125=5 (45)(162025) x xxx

42. 3 7291331 q + 323 2 7291331

Sumofcubes;729=9 and1331=11 (911)(8199121) q qqq + +−+

43. 331251728rs + 3333 22 1251728

Sumofcubes;125=5 and1728=12 (512)(2560144) rs rsrrss + +−+

44. ( ) ( ) 21 334131 xxx−+−

3 4131

( ) ( ) 

FactorouttheGCFofeachexpression (1)43(1) Distributethe3 (1)433 Combineliketerms 1(73) xxx xxx xxx xx −+− −+− −+−

) ( ) 21 33 2 3

45. ( ) ( ) 13 44323523 ccc+−+ ( ) ( )   13 44 1 4 1 4 1 4 323523

FactorouttheGCFoftheexpression (23)35(23) Distributethe-5 (23)(31015) Combineliketerms (23)(715) ccc ccc ccc cc +−+ +−+ +−− +−−

46. ( ) ( ) 14 3331037103 ttt+++ ( ) ( )   14 33 1 3 1 3 1 3 31037103

FactorouttheGCFoftheexpression (103)37(103) Distributethe7 (103)(37021) Combineliketerms (103)(7321) ttt ttt ttt tt +++ +++ +++ ++

47. ( ) ( ) 23 5514252 xxx+++

( ) ( )

 23 55 2 5 2 5 2 5 14252

FactorouttheGCFoftheexpression

(2)145(2) Distributethe5

(2)(14510) Combineliketerms

(2)(1910) xxx xxx xxx xx +++ +++ +++ ++

48. ( ) ( ) 16 5593132313 yyy

16 55

( ) ( )

93132313

Factorout theGCFoftheexpression

(313)92(313) Distributethe-2

(313)(9626) Combineliketerms

(313)(326) yyy yyy yyy yy −−+ −+

49. ( ) ( ) 31 225291129 zzz−+−

( ) ( )   31 22 3 2 3 2 3 2 5291129

Factorout theGCFoftheexpression

(29)511(29) Distributethe11

(29)(52299) Combineliketerms

(29)(2799) zzz zzz zzz zz −+− −+− −+−

50. ( ) ( ) 15 66623523 ddd+++

( ) ( )

15 66 1 6 1 6 1 6 623523

FactorouttheGCFoftheexpression

(23)65(23) Distributethe5

(23)(61015) Combineliketerms

(23)(1615) ddd ddd ddd dd +++ +++ +++ ++

Real-World Applications

For the following exercises, consider this scenario: Charlotte has appointed a chairperson to lead a city beautification project. The first act is to install statues and fountains in one of the city’s parks. The park is a rectangle with an area of

2 9810527 xx+− square meters, as shown. The length and width of the park are perfect factors of the area.

51. Factor by grouping to find the length and width of the park.

2 2 9810527

Rewrite105x as126x-21x (98126)(2127) Factorout theGCFofeachexpression 14(79)3(79) Usethedistributiveproperty (143)(79) xx xxx xxx xx +− ++−− +−+ −+

52. A statue is to be placed in the center of the park. The area of the base of the statue is

2 4129 xx++ . Factor the area to find the lengths of the sides of the statue.

2 2 4129

Rewrite12xas6x+6x (46)(69) FactorouttheGCFofeachexpression 2(23)3(23) Usethedistributiveproperty (23)(23) xx xxx xxx xx ++ +++ +++ ++ 2 Rewrite (23) x +

53. At the northwest corner of the park, the city is going to install a fountain. The area of the base of the fountain is 2 925 x . Factor the area to find the lengths of the sides of the fountain.

(915)(1525) Factorout theGCFofeach expression

3(35)5(35) Usethedistributive property

(35)(35) x

For the following exercise, consider the following scenario: A school is installing a flagpole in the central plaza. The plaza is a square with side length 100 yards as shown. The flagpole will take up a square plot with area 2 69xx−+ square yards.

54. Find the length of the base of the flagpole by factoring.

FactorouttheGCFofeachexpression

(3)3(3) Usethedistributiveproperty

(3)(3) Rewrite (3)

Extensions

For the following exercises, factor the polynomials completely.

55. 42 16200625 xx−+ 42222 422 222 16200625

Rewrite -200 as -100 - 100 (16100)(100625)

4(425)25(425)

Rewrite (425)

Usethedistributiveproperty (4 xxxxx xxx xxx −+ −+−+ 22 22 22 25)(425)

FactorouttheGCFof eachexpression

Rewrite0 as10 - 10 ((410)(1025))

Usethedistributiveproperty ((25)(25))

FactorouttheGCFofeachexpression (2(25)5(25)) xx xxxx xxx xxx ++−− +−+ 2 2 22

Distributetheexponent (25)(25) xx xx −+ −+

56. 4 81256 y 4222 422 222 81256

Rewrite0 as-144 144 (81144)(144256) Factorout theGCFofeachexpression

thedistributiveproperty (916)(916)

Rewrite0 as-12 12 (916)[(912)(1216)]

9(916)16(916) Use yyyy yyy yyy −+ −+− −+− 22 22 2

FactorouttheGCFofeachexpression (916)[3(34)4(34)] yyyyy yyyy yyyy +−+ +−+− +−+− 2

Usethedistributiveproperty (916)(34)(34) yyy++−

57. 44162401za 44222222 422224 22222 162401

Rewrite0z as-196z196 (16196)(1962401)

FactorouttheGCFofeachexpression 4(449)49(449 zaaaza zzazaa zzaaza −+ −+− −+− 2 2222 2222 )

Usethedistributiveproperty (449)(449) Rewrite0zaas -14 14 (449)[(414)(1449)]

FactorouttheGCFofeachexpressio zazazaza zazzazaa +−+ +−+− 22 22 n (449)[2(27)7(27)]

Usethedistributiveproperty (449)(27)(27) zazzaaza zazaza +−+− ++−

58. ( ) ( ) 23 42532128 xxx+++

5(32)4(32)

( ) ( ) 23 42 133 222 1 2 2 532128

Factorout theGCFoftheexpression

Squaretheexpression

(32)[58(9664)] Combineliketerms

Factorout theGCFoftheexpressions; Evaluate (32)[58(32)] xxx xxx xxx +++ +++ +++ 1 2 2 1 2 2 1 2 2

(32)[58(9124)] Distributethe 8 (32)(5729632) Combineliketerms (32 xxxxx xxxx xxxx x +++++ ++++ ++++ + 1 2 2 )(7210132) xx++

59. ( ) 1 32 3248162243 xxx+−− ( ) 1 32 21 21 3248162243 Notethat (49)is afactoroftheexpression ((49)(8627)) Rewrite -6 as -18 12 ((49)((818)(1227))) Factorout theG xxxx xxxxxx xxxx +−−+ +−−+ +−+− 1 1 CFofeach expression ((49)(2(49)3(49)) Usethedistributiveproperty ((49)(23)(49)) Rewrite 1 (49)(49)(23) xxxx xxx xxx +−+− ++− +−+

Chapter 1 Prerequisites

1.6 Rational Expressions

Chapter 1 Review Exercises

Chapter 1 Practice Test

Section Exercises

Verbal

1. How can you use factoring to simplify rational expressions? You can factor the numerator and denominator to see if any of the terms can cancel one another out.

2. How do you use the Least Common Denominator to combine two rational expressions?

You express each denominator as a factor of the LCD and then multiply each side by the other term over itself. This creates a common denominator so the expression can be combined.

3. Tell whether the following statement is true or false and explain why: You only need to find the Least Common Denominator when adding or subtracting rational expressions.

True, multiplication and division do not require finding the LCD because the denominators can be combined through those operations, whereas addition and subtraction require like terms.

Algebraic

For the following exercises, simplify the rational expressions.

Differenceofsquares; rewrite -5 as - - 4 54

(4)(4)

FactorouttheGCFofeachexpression ()(44)

(4)(4) Usethedistributivepro (1)4(1) x xxx xx xx xxx xx xxx −+ −+ −+−+ −+ perty

(4)(4) Simplify (4)(1) 4 1 xx xx x x −+ +

2 1025 1130 yy yy ++ ++

(5)(525)

Factorout theGCFofeachexpression (6)(530)

2 2 2 1025 Rewrite10 as 5 5; Rewrite11 as 6 5 1130

(5)5(5)

Usethedistribu (6)5(6) yy yyyyyy yy yyy yyy yyy yyy ++ ++ ++ +++ +++ +++ +++ tiveproperty

(5)(5) Simplify (5)(6) 5 6 yy yy y y ++ ++ + +

(612)(1224)

FactorouttheGCFofeachexpression (612)(1224)

6(2)12(2)

Usethedistributiveproperty

(612)(2) Simplify (612)(2)

9189 Rewrite18 as 9 9 33 (99)(99)

Factorout theGCFofeachexpression 3(1)

9(1)9(1)

Usethedistributiveproperty 3(1) (99)(1) 3(1) bb bbb b bbb b bbb b bb b ++ + + +++ + +++ + ++ +

Factorout the9

9(1)(1) Simplify 3(1) 3(1)33 bb b bb ++ + +=+

Factorout theGCFofeachexpression (12)(12144)

Usethedistributiveproperty (12)12(12)

(12)(1

(28)(4)

FactorouttheGCFoftheexpressions (44)(22)

2(4)(4)

Usethedistribu 4(1)2(1)

(21)(4)

Factoroutthe2 (42)(1) (21)(4) Simplify 2(21)(1)

10. 2 2 654 31920 xx xx +− ++ 2 2 2 2 654

Rewrite 5 as 8 - 3; Rewrite19 as15 4 31920

(68)(34)

Factorout theGCFofeachexpression (315)(420)

2(34)(34) Usethe 3(5)4(5) xx xxxxxx xx xxx xxx xxx xxx +− + ++ ++−− +++ +−+ +++ distirbutiveproperty (21)(34) Simplify (34)(5) 21 5 xx xx x x −+ ++ +

11. 2 2 918 318 aa aa ++ +− 2 2 2 2 918 Rewrite 9 as 6 3; Rewrite 3aas 6 - 3 318 (6)(318)

FactorouttheGCFoftheexpressions (6)(318)

(6)3(6) Usethedistrib (6)3(6) aa aaaaa aa aaa aaa aaa aaa ++ + +− +++ ++−− +++ +−+ utiveproperty (3)(6) Simplify (3)(6) 3 3 aa aa a a ++ −+ + 12. 2 2 32518 32314 cc cc +− −+

2 2 2 2 32518

Rewrite 25 as 27 - 2 ; Rewrite -23 as -21 - 2 32314 (327)(218)

Factorout theGCFofeachexpression (321)(214)

3(9)2(9) 3(7)2(7)

Usethedistributiveproperty (32)(9)

Simplify (32)(7) 9 7 cc cc c c −+ +

13. 2 2 12298 2853 nn nn 2 2 2 2 12298

Rewrite -29 as 3 - 32n; Rewrite-5 as -12 7 2853 (123)(328)

Factorout theGCFoftheexpressions (2812)(73)

3(41)8(41) 4(73)(73) nn nnnnn nn nnn nnn nnn nnn + ++−− −+− +−+ −+−

Usethedistributiveproperty (38)(41)

Simplify (41)(73) 38 73 nn nn n n −+ +−

For the following exercises, multiply the rational expressions and express the product in simplest form.

14. 22 22 62715 269 xxxx xxx −−+−  +−−

15. 22 22 2241024 1236816 cccc cccc +−−+  ++−+ 22 22 22 22 2241024

Rewritetheexpressions 1236816 (6)(424)(6)(424)

FactorouttheGCFoftheexpressions (6)(636)(4)(416) cccc cccc cccccc cccccc c +−−+  ++−+ ++−−−+−+  +++−+−+ (6)4(6)(6)4(6)

Usethedistributiveproperty (6)6(6)(4)4(4) (4)(6)(4)(6)

Simplify (6)(6)(4)(4) 6 6 ccccc cccccc cccc cccc c c +−+−−− +++−−− −+−−  ++−− +

16. 22 22 29353221 102131449 dddd dddd +−+− +++−

19. 22 22 2152521525 425251 dddd dd ++−+

20. 22 22 65502076 1544202910 xxxx xxxx  −−++

21. 22 22 1215 4343 ttt tttt −+−  ++−+ 22 22 2 22 1215

FactorouttheGCFoftheexpressions (3)(3)(3)(3) (1) ttt tttt ttttt tttttt t −+− ++−+ −+++−− +++−+−+ (1)(5)3(5)

Rewritethe expressions; Differenceofsquares 4343 (1)(1)(5)(315)

Usethedistributiveproperty (3)(3)(3)(3) (1)(1)(3)(5)

Simplify (1)(3)(1)(3)

5 3 tttt tttttt tttt tttt t t ++−+  +++−−− −+−+  ++−− + +

22. 22 22 21512133 61354159 nnnn nnnn −−−+  +−−+

23. 22 22 362533220 66550182710 xxx xxxx −++ ++++

Rewrite theexpressions 66550182710 (65)(65)(330)(220)

22 22 2 22 362533220

Factorout theGCFoftheexpression (65)(6050)(1815)(1210) xxx xxxx xxxxx xxxxxx

++++++ s (65)(65)3(10)2(10)

Usethedistributiveproperty (65)10(65)3(65)2(65) (65)(65)(32)(10)

Simplify (10)(65)(32)(65) 65 65 xxxxx xxxxxx xxxx xxxx x x −++++  ++++++ −+++ ++++ +

For the following exercises, divide the rational expressions. 24. 22

Rewritetheexpressions 23923 (39)(26)(2)(2)

22 22 22 22 3762

FactorouttheGCFofeachexpression; divide (26)(39)(23)(23) 3( yyyy yyyy yyyyyy yyyyyy y −−+−  −−+− −+−++−−  −+−++−− 3)2(3)(23)(23)

Usethedistributiveproperty 2(3)3(3)(2)(2) (32)(3)(1)(23)

Simplify (23)(3)(1)(2) 32 2 yyyyy yyyyyy yyyy yyyy y y −+−+−+ −+−+−+ +−−+  +−−+ + +

25. 22 22 6126114 81892116 pppp pppp +−−+  +++−

26. 22 22 923 6923 qqq qqqq  +++−

22 22 2 22 923

Rewritethe expressions; divide 6923 (3)(3)(3)(3)

FactorouttheGCFoftheexpressions (3)(39)(3)(3) (3)(3) (3)3(3) qqq qqqq qqqqq qqqqqq qq qqq  +++− −+++−−  +++−+− −+ +++ (3)(3)

Usethedistributiveproperty (3)(3) (3)(3)(1)(3)

Simplify (3)(3)(1)(3) 1 1 qqq qqq qqqq qqqq q q +−+ −+− −+−+  +++− +

27. 22 22 18771832944 271529154 dddd dddd +−+−  −+−+

28. 22 22 16185521730 3236114256 xxxx xxxx +−++  −−++

For the following exercises, add and subtract the rational expressions, and then simplify.

33.

410 xy +

410 TheLCDis

410 Multiply

410 Add

+ +

410 xy xy yx xyyx yx xyxy yx xy +

34.

FactorouttheGCFofeachexpression

combinefractions

Multiply

(3)(1)(3)(2) UsetheF (2)(1)(1)(2)

38. 123 121 xx xx −+ ++ 123

TheLCDis ( 1)(2 1) 121 121231

Multiply 121211 (1)(21)(23)(1) Use (1)(21)(21)(1)

Combineliketerms

39. 325 12 zz zz + + +− 325 The LCDis ( 1)( - 2) 12 32251

Multiply 1221 (3)(2)(25)(1)

UsetheFOILmethod (1)(2)(2)(1)

Combineliketerms

40.

UsetheFOILmethod 4(1)4(1)

Combineliketerms

UsetheFOILmethod (1)(1)(1)(1)

For the following exercises, simplify the rational expression.

44.

Rewritetheexpression

Simplifyandmultiply; TheLCDis ( - 1)( (1)(1)(1)(1) xx

Distribute

Multiply 11(1)(1)

47. ab ba ab ab

ab ba ab ab abab abba abab babaab ab abab ab ab

49.

Rewritetheexpression

Distribute 2121 2(1)(1)(1)

Simplify;T (21)(2)(21)(1)

heLCDis (2 1)( 2)

Multiply (21)(2)212

Combineliketerms

Combineliketerm

50. xy yx xy yx +

Rewritethedenominator;theLCDis Multiply Add

Rewritetheexpressionwiththenewdenominator

Rewritetheexpression Distribute

Real-World Applications

51. Brenda is placing tile on her bathroom floor. The area of the floor is 2 1587 xx square feet. The area of one tile is 2 21xx−+ . To find the number of tiles needed, simplify the rational expression: 2 2 1587 21 xx

Rewrite -8 as -15 7; Rewrite -2 as -21 (1515)(77)

2 2 2 2 1587

FactorouttheGCFofeachexpression ()(1)

hedistributiveproperty

15(1)7(1) Uset (1)(1) xx xxxxxx xx xxx

(157)(1) Simplify (1)(1) 157 1 xx xx x x +− +

52. The area of Lijuan yard is 2 25625 x square feet. A patch of sod has an area of 2 1025xx−+ square feet. Divide the two areas and simplify to find how many pieces of sod Lijuan needs to cover her yard.

2 2 2 25625

2 x

Differenceofsquares; Rewrite -10 as -5 - 5 1025 (525)(525)

FactortheGCFofeachexpression 5525

25(5)(5) Usethedistributiveproperty (5)5(5)

5(5)(5) Simplify (5)(5)

25(5) Distribute 5 25125 5 xx xx x x x x −+ + +

53. Elroi wants to mulch his garden. His garden is 2 1881xx++ square feet. One bag of mulch covers 2 81 x square feet. Divide the expressions and simplify to find how many bags of mulch Elroi needs to mulch his garden. 2 2 2 1881

Rewrite18 as 9 9; differenceofsquares 81 (9)(981)

FactorouttheGCFofeachexpression (9)(9) (9)9(9)

Usethedistributiveproperty (9)(9)

(9)(9) Simplify (9)(9) 9 9

Extensions

For the following exercises, perform the given operations and simplify.

54.

Rewritetheexpressions;divide

FactorouttheGCFofeachexpression (3)2(3) (3)(

Usethedistributiveproperty (2)(3)(23)(3)(1)(1) (1)(3)(1)(2)(56)(23) Simplify

13(3)(3)2(4)5(4) 43(2)(2)2(3)5(3)

Usethedistributiveproperty

1(31)(3)(25)(4) 4(31)(2)(25)(3)

Rewritetheexpression

TheLCDis

Multiply

Combineliketerms

Distributeandmultiply

UsetheFOI

Combineliketerms

Rewritetheexpressions; divide 431281612243737 3263127282323

Factorout theGCFofeachexpression (4)3(

4)8(2)12(2)(37)(37) (3)2(3)3(4)7(4)(23)(23)

Usethedistributiveproperty (3)(4)(812)(2)(1)(37) (2)(3)(37)(4)(1)(23)

Simplify 4(23) 4 23

Chapter 1 Review Exercises

Section 1.1 For the following exercises, perform the given operations.

3. 2 2562 +

2 2562 Sqaure the 5

22562 Multiply

5062 Divide

503 Add 53 + + + +

For the following exercises, solve the equation.

4. 5911 x +=−

5911 Subract 9 fromboth sides

520 Divide both sides by5 4 x x x +=− =− =−

5. 2 2464 y += 2 2464 Sqaurethe4 21664 Subtract 16 fromboth sides

248 Divide2 fromboth sides

24 y y y y += += = =

For the following exercises, simplify the expression.

6. ( ) 92321 y ++ ( ) 92321 Distribute (918)321 Divide (36)21 Multiply 6121 Add 613 y y y y y ++ ++ ++ ++ +

7. ( )347mm +− ( )347 Addinsideoftheparenthesis 3(11) Multiply 33 Subtract 32 mm mm mm m +−

For the following exercises, identify the number as rational, irrational, whole, or natural. Choose the most descriptive answer.

8. 11

9. 0

10. 5 6 :rational

11. 11 irrational

Section 1.2

For the following exercises, simplify the expression.

12. 2422

20. Write the number in standard notation:

21. Write the number in scientific notation:

Section 1.3

For the following exercises, find the principal square root.

22.

23.

24.

25.

Section 1.4

For the following exercises, perform the given operations and simplify.

35. ( ) ( ) 32 321427 xxxx +−+−+ ( ) ( )

36.

)

37. ( ) ( ) 22 236349 xxxx +−+−+

( ) ( ) 22 2 236349 53 xxxx xx +−+−+ −+

38. ( ) ( ) 22 6310635 aaaa ++−−+

( ) ( ) 22 22 6310635 6310635 65 aaaa aaaa a ++−−+ ++−+− +

39. ( )( )36kk+−

( )( ) 2 2 36 6318 318 kk kkk kk +− −+−

40. ( )( )2132 hh+−

( )( ) 2 2 2132 6432 62 hh hhh hh +− −+−

41. ( )( ) 2 11xx++

( )( ) 2 32 11 1 xx xxx ++ +++

42. ( )( ) 2 223mmm−+−

( )( ) 2 322 3 223 23246 76 mmm mmmmm mm −+− +−−−+ −+

43. ( )( ) 23 abab +−

( )( ) 22 22 23 362 352 abab aababb aabb +− −+− +−

44. ( )( )xyxy +− ( )( ) 22 22 xyxy xxyxyy xy +− −+−

Section 1.5

For the following exercises, find the greatest common factor.

45. 22 81927ppqpq +− 9 p

46. 22 12418xyxyxy +− 2xy

47. 322884144 ababa +−

2 4a

For the following exercises, factor the polynomial.

48. 2 2918 xx 2 2 2918

Rewrite -9 as -12 3 (212)(318) Factorout theGCFofeachexpression

2(6)3(6) Usethedistributiveproperty (23)(6) xxxxx xxx xxx xx −−+ −+− −+− +−

49. 2 83027 aa+− 2 2 83027

Rewrite 30 as 36 - 6 (836)(627) Factorout theGCFofeachexpression

4(29)3(29) Usethedistributive property (43)(29) aaaaa aaa aaa aa +− ++−− +−+ −+

50. 2 566dd

2 2 566

Rewrite -5 as -11 6 (11)(666) Factorout theGCFofeachexpression (11)6(11) Usethedistributiveproperty (6)(11) ddddd ddd ddd dd −−+ −+− −+− +−

51. 2 1025xx++

2 2 2 1025 Rewrite10 as5 5 (5)(525) FactorouttheGCFofeachexpression (5)5(5) Usethedistributiveproperty (5)(5) Rewrite (5) xxxxx xxx xxx xx x +++ +++ +++ ++ +

52. 2 69yy−+

2 2 2 69 Rewrite -6 as -3 - 3 (3)(39) FactorouttheGCFofeachexpression (3)3(3) Usethedistributiveproperty (3)(3) Rewrite (3) yyyyy yyy yyy yy y −+ −+−+

53. 224129 hhkk −+ 22 22 4129 Rewrite -12 as -6 - 6 4669 FactorouttheGCFofeachexpression 2(23)3(23) Usethedistributiveproperty (23)(23) hhkkhkhkhk hhkhkk hhkkhk hkhk −+ −−+ 2 Rewrite (23) hk

54. 2 361121 x 2 2 361121

Rewrite 0 = 209 - 209 (361209)(209121) Factorout theGCF ofeachexpression 19(1911)11(1911) Usethedistributiveproperty (1 xxxx xxx xxx ++−− +−+ 911)(1911) xx−+

55. 3 216 p + 3 2 216 Sumofcubes (6)(636) p ppp + +−+

56. 3 8125 x 3 2 8125 Differenceofcubes (25)(41025) x xxx−++

57. 336427qp 33 22 6427 Differenceofcubes (43)(16129) qp qpqpqp −++

58. ( ) ( ) 13 444131 xxx−+− ( ) ( )

(1)43(1) Distributethe3

(1)(433) Combineliketerms

(1)(73) xxx xxx xxx xx −+− −+− −+−

59. ( ) ( ) 14 333383 ppp+−+ ( ) ( )   14 33 1 3 1 3 1 3 3383 FactorouttheGCFoftheexpression (3)38(3) Distributethe-8 (3)(3824) Combineliketerms

(3)(524) ppp ppp ppp pp +−+ +−+ +−− +−−

60. ( ) ( ) 21 33421521 rrr ( ) ( )

421521

FactorouttheGCFoftheexpression

(21)45(21) Distributethe-5 (21)(4105) Combineliketerms

(21)(65)

Section 1.6

For the following exercises, simplify the expression.

(4)(312)

FactorouttheGCFofeachexpression (4)(416) (4)3(4) Usethedis (4)4(4)

tributiveproperty (3)(4) Simplify (4)(4) 3

62. 2 2 425 42025 y yy−+ 2 2 2 2 425

Rewrite 0 as10 - 10; rewrite -20yas -10 - 10 42025

(410)(1025)

FactorouttheGCFofeachexpression (410)(1025)

2(25)5(25)

2(25)5(25) y xyyyy yy yyy yyy yyy yyy −+ ++−− −+−+ +−+

(25)(25)

Usethedistributiveproperty

Simplify (25)(25) 25 25 yy yy y y −+ +

63. 22 22 235194 26810133 aaaa aaaa

2a 2 - a - 3

2a 2 - 6a - 8 × 5a 2 -19a - 4 10a 2 -13a - 3

(2a 2 + 2a) + (-3a - 3)

(2a 2 + 2a) + (-8a - 8) × (5a 2 - 20a) + (a - 4) (10a 2 -15a) + (2a - 3)

2a(a +1) - 3(a +1)

2a(a +1) - 8(a +1) × 5a(a - 4) + (a - 4) 5a(2a - 3) + (2a - 3)

(2a - 3)(a +1)

(2a - 8)(a +1) × (5a +1)(a - 4) (5a +1)(2a - 3)

2(a - 4) = 1 2

Rewrite the expressions

Factor out the GCF of each expression

Use the distributive property

Simplify a - 4

64. 22 43 916 dd dd 

65. 22 22 56239 253443 mmmm mmmm

66. 22 22 472861 617106710 dddd dddd −−++  −++−

67. 106 xy +

106 Add 610 xy yx xyyx yx xyxy xy xy + + + +

106 Multiply

68.

Rewritetheexpressions

FactorouttheGCFfromeachexpression 1(1)(1)

Usethedistributiveproperty

(1)(1)(1)(1)

TheLCDis (1)( 1)(-1) (1)(1)(1)(1)

Multiply (1)(1)1(1)(1)1 121233

(1)(1)(1)(1)(

Chapter 1 Practice Test

For the following exercises, identify the number as rational, irrational, whole, or natural. Choose the most descriptive answer.

1. 13 rational 2. 2 irrational

For the following exercises, evaluate the expression.

3. ( ) 2312;2 xx+−= 2(23)12 Substitute 2 2(5)12 Combine 1012 Distributethe2 2 Combine x +−= =−

4. ( )2 3326;1yy+−= ( )2 2 13326 Substitute 1 (6)26 Combine 3626 Expand 10 Combine y +−= =

5. Write the number in standard notation: 63.141510  3,141,500

6. Write the number in scientific notation: 0.0000000212. 8 2.1210  For the following exercises, simplify the expression.

8. ( ) ( )4362 xx+−+ ( ) ( )4362 Distributethe4 and the minus symbol 41262 Combinelike terms

17. 624754126 +−

624754126 24 has factors 4 and 6; 54 has factors 9 and 6 646796126 Multiplication propertyof radicals

646796126 4 and 9 are perfect squares

6(2)67(3)6126 Multiply

18.

19. ( ) ( ) 322 1323653 qqqq +−−+−

( ) ( ) 322 322 32 1323653Distribute the minus symbol 1323653Combine like terms 1345 qqqq qqqq qqq +−−+− +−−−+

20. ( ) ( ) 22 62191 ppp+++−

( ) ( ) 22 2 62191 152 ppp pp +++− +

21. ( )( ) 2 244nnn−−+

( )( ) 2 322 32 244 Usethedistributiveproperty 44288 Combineliketerms 6128 nnn nnnnn nnn −−+ −+−+− −+−

22. ( )( ) 22 abab −+ ( )( ) 22 22 22 UsetheFOILmethod 242 Combineliketerms 232 abab aababb aabb −+ +−− For the following exercises, factor the polynomial.

23. 2 1681 x 2 1681Difference of squares (49)(49) x xx−+

24. 2 1236yy++ 2 2 2 1236 Rewrite12yas6y+6y 6636

(6)(6) Rewrite (6) yy yyy yyy yy y ++ +++ +++ ++ +

FactorouttheGCFofeachexpression

(6)6(6) Usethedistributiveproperty

25. 3 271331 c 3 2 271331 Differenceofcubes (311)(933121) c ccc −++

26. ( ) ( ) 13 443626 xxx−+− ( ) ( )

3626

13 44

FactorouttheGCFoftheexpression (6)32(6) Distributethe2

(6)(3212) Combineliketerms

(6)(512) xxx xxx xxx xx −+− −+− −+−

For the following exercises, simplify the expression. 27. 22 22 2734159 941 zzzz zz ++−+