12-2:SIMPLIFYINGRATIONALEXPRESSIONS

LessonObjectives:• Simplifyingrationalexpressions• Factoringcompletely

ARATIONALFUNCTIONcanbewrittenintheform f x( ) = polynomial

polynomial.Thevalueofthevariablecannotmakethe

denominatorequalto0.EXAMPLE1(12-1):GRAPHINGRATIONALFUNCTIONS

1. Graph y = 1x

2. Graph y = 4x +3

+ 2

ARATIONALEXPRESSIONisanexpressionwhichcanbewrittenintheformpolynomial

polynomial.Ofcourse,thevalueofthe

expressioninthedenominatorcannotbezero,sincedivisionbyzeroisundefined.OntheKeystoneExam,thesevaluesareusuallygivenintheproblem.Youwon’thavetoworryaboutfindingthevaluesonyourown.Likerationalnumbers,arationalexpressionisinsimplestformifthenumeratoranddenominatorhavenocommonfactorsexcept1.EXAMPLE2:SIMPLIFYINGARATIONALEXPRESSIONSimplify.

3.6x +12x + 2

4.6a+ 912

5.4x3

28x4 −8x2 6.

2m− 56m−15

7.2p− 244p− 48

8.3x2 − 9xx −3

9.3x + 63x2

10.20+ 4tt + 5

11.2x −12

x2 − 7x + 6 12.

2b−8b2 −16

13.m+ 6

m2 −m− 42 14.

w2 + 7ww2 − 49

15.a2 + 2a+15a+ 5

16.m2 + 7m+12m2 + 6m+8

17.c2 − 6c+8c2 + c− 6

18.b2 +8b+15

b+ 5

Thenumeratoranddenominatorofx − 3

3− xareopposites.Tosimplifytheexpression,youcanfactorouta-1from3–xtoget

−1 3− x( ) ,whichyoucanrewriteas−1 x − 3( ) .Thensimplify x − 3

−1 x − 3( ).

EXAMPLE3:RECOGNIZINGOPPOSITEFACTORSSimplify.

19.5x −159− x2

20.5− 4n4n− 5

21.12− 4tt2 − 2t −3

22.4m−84− 2m

23.m− 24− 2m

24.v− 525− v2

25.4−w

w2 −8w+16 26.

−2x2 + 20x − 4249− x2

EXAMPLE4:EVALUATINGARATIONALEXPRESSION27.Bakingtimeforaloadofbreaddepends,inpart,onitssizeandshape.Agoodapproximationforthebakingsizeofa

cylindricalloafis60 • volume

surface _ area.Findthebakingtimeforaloafthatis5in.longandhasaradiusof4in.

28.Tokeepheatingcostsdownforastructure,architectswanttheratioofsurfaceareatovolumeassmallaspossible.Findanexpressionfortheratioofthesurfaceareatovolumeforeachshape.a)squareprism b)cylinder

EXAMPLE5(12-3):MULTIPLYINGANDDIVIDINGRATIONALEXPRESSIONSSimplify.

29.m− 23m+ 9

•2m+ 62m− 4

30.t2 + 5t + 6t −3

•t2 − 2t −3t2 +3t + 2

31.x2 − 25x2 − 4x⎛

⎝⎜

⎞

⎠⎟

x2 + x − 20x2 +10x + 25⎛

⎝⎜

⎞

⎠⎟ 32.

7t2 − 28t2t2 − 5t −12

•6t2 − t −1549t3

33.5x2 +10x −155− 6x + x2

÷2x2 + 7x +34x2 −8x − 5

34.x2 + x − 6x2 − x − 6

÷x2 + 5x + 6x2 + 4x + 4

675

Lesson Quiz

Simplify each expression.

1. –2

2.

3.

4.

5.

Alternative Assessment

Write rational expressions onindividual index cards. Makeenough for half of the class. Write the simplifications of therational expressions on otherindex cards for the other half ofthe class. Have students find“matching” cards. Shuffle thecards and repeat.

Test Prep

ResourcesFor additional practice with avariety of test item formats:• Standardized Test Prep, p. 719• Test-Taking Strategies, p. 714• Test-Taking Strategies with

Transparencies

4x 1 1x2

4x2 1 xx3

3x 1 44x 1 1

6x2 2 x 2 128x2 2 10x 2 3

21x 1 6

x 2 636 2 x2

xx 2 8

x2 1 8xx2 2 64

6 2 2xx 2 3

PowerPoint

4. Assess & Reteach

Lesson 12-2 Simplifying Rational Expressions 675

35. a. Construction To keep heating costs down for a structure, architects want theratio of surface area to volume as small as possible. Find an expression forthe ratio of the surface area to volume for each shape.i. square prism ii. cylinder

b. Find the ratio for each figure when b = 12 ft, h = 18 ft, and r = 6 ft. ;

36. Error Analysis Explain what error the student made in simplifying the rational expression at the right. See left.

37. Writing Explain why is not the same as x - 3.

Probability If a point is selected at random from a figure and is equally likely to beany point in the figure, then the probability that the point is in a shaded part of the

figure is . Find the probability that a point will be in the shaded

part of each figure.

38. 39.

40. 41.

Simplify each expression. 42–44. See left.

42. 43. 44.

Reasoning Determine whether each statement is sometimes, always, or never truefor real numbers a and b.

45. = 2 sometimes 46. = ab sometimes 47. =

Standardized Test Prep

48. Which expression simplifies to -1? C

A. B. C. D.4 2 p4 1 p

n 2 22 2 n

r 1 33 2 r

x 1 1x 2 1

Multiple Choice

a 1 52

a2 1 6a 2 52a 1 2

ab3

b42bb

36v2 2 49w2

18v2 1 9vw 2 14w2a2 2 5ab 1 6b2

a2 1 2ab 2 8b2m2 2 n2

m2 1 11mn 1 10n2

ChallengeCC

2t t !6

6

4

3t

3y4(y 1 4)

3y8

2y8

14

3x

6x

5w5w 1 6

6 5w

5w

6

area of shaded partarea of whole figure

x2 2 9x 1 3

x2 ! 2x2x

x2 ! 2x2x

x2

"

"

49

49

r

hh

bb

2h 1 2rrh

2b 1 4hbh

ConnectionReal-World

A cylinder with a height of500 ft and a volume of 1 3 106 ft3 has about 89% ofthe surface area of a squareprism with the same heightand volume.

Test Prep

36. The student canceledterms instead of factors.

t 1 33(t 1 2)

never

–3 is not in the domain of .x2 2 9x 1 3

42.

43.

44. 6v 2 7w3v 2 2w

a 2 3ba 1 4b

m 2 nm 1 10n

GO nlineHomework HelpVisit: PHSchool.comWeb Code: ate-1202

lesson quiz, PHSchool.com, Web Code: ata-1202

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675

Lesson Quiz

Simplify each expression.

1. –2

2.

3.

4.

5.

Alternative Assessment

Write rational expressions onindividual index cards. Makeenough for half of the class. Write the simplifications of therational expressions on otherindex cards for the other half ofthe class. Have students find“matching” cards. Shuffle thecards and repeat.

Test Prep

ResourcesFor additional practice with avariety of test item formats:• Standardized Test Prep, p. 719• Test-Taking Strategies, p. 714• Test-Taking Strategies with

Transparencies

4x 1 1x2

4x2 1 xx3

3x 1 44x 1 1

6x2 2 x 2 128x2 2 10x 2 3

21x 1 6

x 2 636 2 x2

xx 2 8

x2 1 8xx2 2 64

6 2 2xx 2 3

PowerPoint

4. Assess & Reteach

Lesson 12-2 Simplifying Rational Expressions 675

35. a. Construction To keep heating costs down for a structure, architects want theratio of surface area to volume as small as possible. Find an expression forthe ratio of the surface area to volume for each shape.i. square prism ii. cylinder

b. Find the ratio for each figure when b = 12 ft, h = 18 ft, and r = 6 ft. ;

36. Error Analysis Explain what error the student made in simplifying the rational expression at the right. See left.

37. Writing Explain why is not the same as x - 3.

Probability If a point is selected at random from a figure and is equally likely to beany point in the figure, then the probability that the point is in a shaded part of the

figure is . Find the probability that a point will be in the shaded

part of each figure.

38. 39.

40. 41.

Simplify each expression. 42–44. See left.

42. 43. 44.

Reasoning Determine whether each statement is sometimes, always, or never truefor real numbers a and b.

45. = 2 sometimes 46. = ab sometimes 47. =

Standardized Test Prep

48. Which expression simplifies to -1? C

A. B. C. D.4 2 p4 1 p

n 2 22 2 n

r 1 33 2 r

x 1 1x 2 1

Multiple Choice

a 1 52

a2 1 6a 2 52a 1 2

ab3

b42bb

36v2 2 49w2

18v2 1 9vw 2 14w2a2 2 5ab 1 6b2

a2 1 2ab 2 8b2m2 2 n2

m2 1 11mn 1 10n2

ChallengeCC

2t t !6

6

4

3t

3y4(y 1 4)

3y8

2y8

14

3x

6x

5w5w 1 6

6 5w

5w

6

area of shaded partarea of whole figure

x2 2 9x 1 3

x2 ! 2x2x

x2 ! 2x2x

x2

"

"

49

49

r

hh

bb

2h 1 2rrh

2b 1 4hbh

ConnectionReal-World

A cylinder with a height of500 ft and a volume of 1 3 106 ft3 has about 89% ofthe surface area of a squareprism with the same heightand volume.

Test Prep

36. The student canceledterms instead of factors.

t 1 33(t 1 2)

never

–3 is not in the domain of .x2 2 9x 1 3

42.

43.

44. 6v 2 7w3v 2 2w

a 2 3ba 1 4b

m 2 nm 1 10n

GO nlineHomework HelpVisit: PHSchool.comWeb Code: ate-1202

lesson quiz, PHSchool.com, Web Code: ata-1202

GPS