Worksheet 2 3 Algebraic Fractions - Macquarie University

1 Worksheet2:3 AlgebraicFractionsSection1 FactoringandAlgebraicFractionsAspointedo utin worksheet2:1, we canusefactoringto simplifyalgebraicexpressions,andinpartic ularwe canuseit to erentdenominators,we must rst :a+b2 2a5=5(a+b)5 2 2(2a)2 5=5a+ 5b10 4a10=5a+ 5b 4a10=a+ 5b10 Example2:32x y+ 13xy=3y 33y 2x 2(y+ 1)2 3xy=9y6xy 2y+ 26xy=9y (2y+ 2)6xy=9y 2y 26xy=7y 26xyExample3:2y+ 1+3y 1=2(y 1)(y+ 1)(y 1) 3(y+ 1)(y 1)(y+ 1)=2y 2(y+ 1)(y 1) 3y+ 3(y 1)(y+ 1)=2y 2 3y 3(y+ 1)(y 1)= y 5(y+ 1)(y 1)= (y+ 5)(y+ 1)(y 1)Sometimesit is di cultto nda simpleexpressionthatis a multipleof two thecaseit is perfectlyacceptableto multiplythetwo theendof thecalculationin the nalfractionthattherearenocommonfactorsin thenumeratoranddenominator;if thereare,youcanalways cancelthemto give :1.

2 Simplifythefollowingalgebraicexpressions :(a)x3+x2(b)m7 m5(c)4t5+t2(d)m+13 m 24(e)3m+47+m 12(f)yy+1 yy+3(g)5t+1+4t 3(h)3mm+4+4mm+5(i)4y+1 5y+2(j)74x+25xySection2 MultiplicationandDivisionAsin numericalfractions,thetrick withsimplifyingthemultiplicationanddivis ionof algebraicfractionsis to look fractionin thatdividingPage2by a fractionis thesameoperationas multiplyingby 1 1x= 1 x1=xFor example116meanshow many 6thsarein onewhole?Theanswer is ,analgebraicexpressionin thenumeratoror denominatorshouldbe treatedas if it werein instance x+ 24= (x+ 2)4= x 24 Example1:x2 x8=x2 8x=8x2x=4 Example2:8x+ 2 72x+ 4=8x+ 2 2x+ 47=8 2(x+ 2)7(x+ 2)=167=227 Example3:x+14yx+48y=x+ 14y x+ 48y=x+ 14y 8yx+ 4=8y(x+ 1)4y(x+ 4)=2(x+ 1)x+ 4 Page3 Example4:4xyx+26=4xy 6x+ 2=24xy(x+ 2)Example5:3xy y6=63x 6y 6y63 2=12xExample6:(xx 1 yy 1)3xy=((y 1)x(y 1)(x 1) (x 1)y(x 1)(y 1))3xy=(xy x xy+y(x 1)(y 1))3xy= x+y(x 1)(y 1) xy3=(y x)xy3(x 1)(y 1)Exercises:1.

3 Simplifythefollowingalgebraicexpressions :(a)m16 5m12(b)3m8 15m20(c)6x+38 2x+112(d)9xy7 6x3(e)6pq5 12p7(f)3(x+1)8 5(x+1)16(g)4x76xy5(h)m+12 m 13m+15(i)3pq 4pp+1(j)8(x+3)9 12(x+1)4(x+3)Page4 Section3 SolvingEquationsSometimeswe areaskedtosolve anequationfora ononesideof anequality signandtheotherinformationin theequationshouldbe similarto solvingequationsin onevariableas inWorksheet2:2. However,youmay endupwithanalgebraicexpressionononesidei nvolvingothervariablesratherthanjusta thesequestionsin thesamewayas thetechniquesgiven in sectionsoneandtwoof :x 23+x+ 15= 3 Multiplyeach sideby 15 - thiswilleliminatethefractions:15 (x 23+x+ 15) =15 315x 23+ 15x+ 13=455(x 2) + 3(x+ 1)=455x 10 + 3x+ 3=458x 7=458x=52x=528=132 Page5 Example2:Solve forxin termsof + 2xyy+ 1=3x(y+ 1) (1 + 2xy)(y+ 1)=(y+ 1)3x(multiplyingbothsidesby (y+ 1))1 + 2xy=3xy+ 3x1=xy+ 3x1=x(y+ 3)(factoringto separatethex)x=1y+ 3 Example3:Solve foryin +x3y2=x+ 12y6y2(4 +x)3y2=6y2(x+ 1)2y(multiplybothsidesby 6y2)2(4+x) =3y(x+ 1)2(4+x)(x+ 1)=3yisolateyy=2(4+x)3(x+ 1)In thelasttwo steps,we wereaimingto makexthesubjectof :1.

4 Solve forx:(a)x+85 x 23= 4(b)x+13+x 42= 5(c)3(x 2)4 2(x+1)5=110(d)4x+1+3x 4=2x+1(e)5x+3+22x+6= 4 Page62. Solve forxin termsofy:(a)3xy= 8(b)4x+1=3y+2(c)4(y+ 1) 3(x+ 5) = 8(d)1+y2+x= 3y(e)5xy+ 3xy2= Simplifythefollowing:(a)x3+x4(b)22xy+4xy 3(c)3x+12 (6x+ 5)(d)3b 1 4b 2(e)2x+2yx+1xy(f)2x2 4x+4x 4(g)1x+12 1(h)1x+1 1x+2+1x+3(i)4a7+3a+52 3(a+2)4(j)3p12 (p2 p4+5p6)2. Simplifythefollwing:(a)4(x+1)3 5(x 2)2(b)x2+3xx+4 2x+85x(c)8x 244 x+712(d)y2 6yy+5 3y+152y 12(e)5m 74m+8 m+23m+6(f)6p 34 4p+ (a)5x6(b) 2m35(c)13t10(d)m+1012(e)13m+114(f)2y(y+1 )(y+3)(g)9t 11(t+1)(t 3)(h)m(7m+31)(m+4)(m+5)(i) y+3)(y+1)(y+2)(j)35y+820xySection21.(a)3 20(b)12(c)412(d)9y14(e)7q10(f) 115(g)1021y(h)5(m+5)6(m+1)(i)12q(p+1)(j) 8(x+1)3 Section31.(a)x= 13(b)x= 8(c)x= 557(d)x= 1(e)x= 1122.

5 (a)x=83y(b)x=4y+53(c)x=4y 193(d)x=1 5y3y(e)x=7y(5+3y) (a)7x12(b)y2+4xy3(c) 9(x+1)2(d) (b+2)(b 1)(b 2)(e)2x(f)2(1+2x)x(x 4)(g)1x+1(h)x2+4x+5(x+1)(x+2)(x+3)(i)37a +2828(j) 5p62.(a) 7x+386(b)2(x+3)5(c)24(x 3)x+7(d)3y2(e)3(5m 7)4(m+2)(f) 92 Page9