Download - 12-2: SIMPLIFYING RATIONAL EXPRESSIONS
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
GPS
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