,,,uebc algebra sample,



           ,special ,symbols



    ,type ,9dicators

,,, "<#f1 #f1 #f"> 2g9 capitaliz$

  passage

, "<#f"> 5d capitaliz$ passage

.1 italic ^w

.7 2g9 italic passage

. 5d italic passage



    ,3/ruc;n ,symbols

( 2g9 frac;n

) 5d frac;n

./ frac;n l9e

< 2g9 ma? gr\p

> 5d ma? gr\p

9 "<#ce"> sup]script



    ,pr9t ,signs

"< op5 p>5!sis

"> close p>5!sis

.< op5 bracket

.> close bracket




       ,special ,symbols "<3t4">

"6 "<#e1 #bce"> plus sign

"- "<#e1 #cf"> m9us sign

"4 "<#e1 #bef"> c5t]$ dot

"7 "<#e1 #bcef"> equal sign

"7@: n equal sign

_= equival5t sign

_6 plus or m9us sign

_- m9us or plus sign

_/ sla%

"9 "<#e1 #ce"> a/]isk
































        ,,,uebc algebra sample,      #dg



444



       #c-#d ,,,special products,



  ,"! >e c]ta9 special products : o3ur s

frequ5tly 9 algebra t !y h be5

classifi$4 ,^! >e giv5 2l4 .7,! lrs 9 !

=mulas may /& = any algebraic expres.n4.

,ea* is a direct result ( ! axioms 9

,*apt] #b4 ,! r1d] %d n only v]ify ea*

by actu,y c>ry+ \ ! /eps & giv+ !

r1sons1 b al memorize !m1 s t he c

recognize bo? ! product f ! factors & !

factors f ! product4

  ;;;"<#c-#aa"> a"<x "6 y"> _= ax "6 ay4

  "<#c-#ab"> "<x "6 y">"<x "- y">

    _= x9#b "- y9#b4

  "<#c-#ac">"9;

        "9 ,! sign _6 is r1d 8plus or

      m9us40 ,if ! upp] "<l[]"> sign is

      us$ 9 ! left memb]1 x is al us$ 9

      ! "r1 s t "<;x _6 y">;9#b

      _= x;9#b _6 #bxy "6 y;9#b       #a




      m1ns "<;x "6 y">;9#b          a#dg

      _= x;9#b "6 #bxy "6 y;9#b &

      "<;x "- y">;9#b

      _= x;9#b "- #bxy "6 y;9#b4

    ;;;"<x _6 y">9#b

    _= x9#b _6 #bxy "6 y9#b4

  "<#c-#ad"> "<x "6 a">"<x "6 b">

    _= x9#b "6 "<a "6 b">x "6 ab4

  "<#c-#ae"> "<ax "6 b">"<cx "6 d">

    _= acx9#b "6 "<ad "6 bc">x "6 bd4

  "<#c-#af"> "<x _6 y">9#c

    _= x9#c _6 #cx9#by "6 #cxy9#b

    _6 y9#c4

  "<#c-#ag"> "<x _6 y">"<x9#b _- xy

    "6 y9#b">

    _= x9#c _6 y9#c4;

  ,! r1d] %d det]m9e : ( ! abv =mulas is

us$ 9 ! foll[+ illu/r,ns4

-------------------------------------#dh

  ,,illu/r,n #a4

  ;;;"<#bx9#b "- #cy">"<#bx9#b "6 #cy">

    _= "<#bx9#b">9#b "- "<#cy">9#b

    _= #dx9#d "- #iy9#b4;



                                      #b




  ,,illu/r,n #b4                    a#dh

  ;;;"<x "6 #b">"<x "6 #e">

    _= x9#b "6 "<#b "6 #e">x "6 #aj

    _= x9#b "6 #gx "6 #aj4;



  ,,illu/r,n #c4

  ;;;"<#cx "6 #dy">"<#bx "- #cy">

    _= #fx9#b "6 "<"-#i "6 #h">xy

      "- #aby9#b

    _= #fx9#b "- xy "- #aby9#b4;



  ,,illu/r,n #d4

  ;;;"<x "6 y "- #a">9#c

    _= .<"<x "6 y"> "- #a.>9#c

    _= "<x "6 y">9#c "- #c"<x "6 y">9#b

      "6 #c"<x "6 y"> "- #a

    _= x9#c "6 #cx9#by "6 #cxy9#b

      "6 y9#c "- #cx9#b "- #fxy

      "- #cy9#b "6 #cx "6 #cy "- #a4;

,"h "<;x "6 ;y"> is 3sid]$ f/ z "o t]m4



  ,,illu/r,n #e

  ;;;"<#cx "6 #by">

      "<#ix9#b "- #fxy "6 #dy9#b">

    _= "<#cx "6 #by">                 #c




      .<"<#cx">9#b                  b#dh

      "- "<#cx">"<#by"> "6 "<#by">9#b.>

    _= "<#cx">9#c "6 "<#by">9#c

    _= #bgx9#c "6 #hy9#c4;



               ,,problems



    ,f9d ! foll[+ products4

;;;#a4 #b;a"<#cx "- #dy">

#b4 "-#cx"<#bx "6 #gy">

#c4 "-#gxy"<#cx9#b "6 #dy">

#d4 #dx9#byz"<z9#b "6 xy "6 yz">

#e4 "<#bx "- #cy">"<#bx "6 #cy">

#f4 "<#gx "6 #ey9#b">"<#gx "- #ey9#b">

#g4 "<x "6 #by">"<x "- #by">

  "<x9#b "6 #dy9#b">

#h4 "<x "- #c">9#b

#i4 "<#bx "6 #gy">9#b

#aj4 "<#cx9#by "- #ez9#b">9#b

#aa4 "<x "- #b">"<x "- #e">

#ab4 "<#bx "6 #c">"<x "- #e">

#ac4 "<xy9#b "- z9#bw">9#b

#ad4 "<#a/bx "6 #b/cy">9#b

#ae4 "<#dx "- #cy">"<#gx "6 #cy">

                                      #d




#af4 .<"<x "6 #a"> "- z.>           c#dh

  .<"<x "6 #a"> "6 z.>

#ag4 "<#bx "6 #cy "6 #c">

  "<#bx "6 #cy "- #c">

#ah4 "<#bx "6 #cy "6 #dz">9#b

#ai4 "<x "- #by "- z">9#b

#bj4 "<#b;a "6 b">9#c

#ba4 "<x "6 #b">"<x9#b "- #bx "6 #d">

#bb4 "<x "- #c">"<x9#b "6 #cx "6 #i">

#bc4 "<x "6 #cy "6 #bz "- #dw">

  "<x "6 #cy "- #bz "6 #dw">

#bd4 "<#dx "- #by "- #cz "6 #cw">

  "<#dx "6 #by "6 #cz "6 #cw">

#be4 "<a "- b "6 c "- d">9#b

#bf4 "<#b;a "6 #c;b "- c "- #d;d">9#b

#bg4 .<#b"<x "6 #by"> "- #c.>

  .<#b"<x "6 #by"> "6 #d.>

#bh4 .<#b"<x "- #cy"> "6 #e.>

  .<#c"<x "- #cy"> "- #b.>

#bi4 "<#bx "6 #cy">9#c

#cj4 "<#ex "- #cy">9#c;

-------------------------------------#di

      #c-#e ,,,factors & factor+,



  ,! process ( factor+ an algebraic   #e




expres.n is simil> to t ( f9d+ !    a#di

factors ( a composite numb]4 ,recall !

4cus.n ( prime & composite 9teg]s 9

,>ticle #a-#d4 ,? process1 : is usu,y

re/rict$ at ? ele;t>y /age to factor+

polynomials ) r,nal coe6ici5ts & to

factors completely free f irr,nal

numb]s1 is frequ5tly p]=m$ by rev]s+ !

processes 3sid]$ 9 ,>ticle #c-#d4 ,s* a

factoriz,n is 3sid]$ complete :5 ea*

algebraic factor is a .7prime factor2. t

is1 an algebraic expres.n t _c 2 factor$

)\t violat+ ! abv re/ric;ns4

  ,! m common types ( factor+ >e

illu/rat$ 2l4 ,note ! import.e &

applic,n ( ! 4tributive axioms 9 ?

4cus.n4



  ,,example #a4 ,factor

#b;ax9#b "- #d;ay9#b "6 #h;a9#bx4

  .1,solu;n4 ,! polynomial 9 ? problem

has #b;a z a common factor4

  ;;;#b;ax9#b "- #d;ay9#b "6 #h;a9#bx

    _= #b;a"<x9#b "- #by9#b "6 #d;ax">4;

                                      #f




  ,,example #b4 ,factor             b#di

x"<a "6 #b;b"> "- #cy"<a "6 #b;b">4

  .1,solu;n4 ,ea* ( ! two expres.ns has

! common t]m "<a "6 #b;b">4 ,"!=e1

  ;;;x"<a "6 #b;b"> "- #cy"<a "6 #b;b">

    _= "<x "- #cy">"<a "6 #b;b">4;



  ,,example #c4 ,factor

"<#dx9#b_/y9#b"> "- "<#i;a "- b">;9#b4

  .1,solu;n4 ,? expres.n is ! di6];e 2t

two p]fect squ>es4

  ;;;(#dx9#b./y9#b) "- "<#i;a "- b">9#b

    _= "<(#bx./y)">9#b

      "- "<#i;a "- b">9#b

    _= .<(#bx./y) "6 "<#i;a "- b">.>

      .<(#bx./y) "- "<#i;a "- b">.>

    _= "<(#bx./y) "6 #i;a "- b">

      "<(#bx./y) "- #i;a "6 b">4;



  ,,example #d4 ,factor

#ix9#b "- #cjxy "6 #bey9#b4

  .1,solu;n4 ,? algebraic expres.n is a

p]fect squ>e4

  ;;;#ix9#b "- #cjxy "6 #bey9#b

    _= "<#cx "- #ey">9#b4;            #g




  ,,example #e4 ,factor             c#di

#bgx9#c "6 "<#h_/y9#c">4

  .1,solu;n4 ,! algebraic expres.n is !

sum ( two cubes4 ,acly1

  ;;;#bgx9#c "6 (#h./y9#c)

    _= "<#cx "6 (#b./y)">

    "<#ix9#b "- (#fx./y)

    "6 (#d./y9#b)">4;

-------------------------------------#ej

  ,,example #f4 ,factor

#abx9#b "6 #gxy "- #ajy9#b4

  .1,solu;n4 ,? trinomial 9 ! =m ( ,eq4

"<#c-#ae"> is factor$ by trial & ]ror4

,! result w 2 9 ! =m

"<ax "6 by">"<cx "6 dy">1 ":

;ac "7 #ab1 bd "7 "-#aj1 &

ad "6 bc "7 #g4 ,"h a & ;c >e bo? plus1

& ;b & ;d >e di6]5t 9 sign4 ,!

correct comb9,n1 we f9d is

#abx9#b "6 #gxy "- #ajy9#b

_= "<#dx "6 #ey">"<#cx "- #by">4



  ,,example #g4 ,factor

#fx9#d "6 #gx9#by9#b "- #cy9#d4

  .1,solu;n4 ,? is ! same type z      #h




,example #f4                        a#ej

  ;;;#fx9#d "6 #gx9#by9#b "- #cy9#d

    _= "<#cx9#b "- y9#b">

    "<#bx9#b "6 #cy9#b">4;



  ,al? ! f/ factor on ! "r is ! di6];e (

two squ>es1 x _c 2 factor$ fur!r1 = s*

factoriz,n wd 9troduce irr,nal

quantities4



               ,,problems



    ,factor ! foll[+ completely4

;;;#a4 #dx "- #bj

#b4 #ajx "6 #aeyz

#c4 #cy9#b "- #iy

#d4 #dx9#cy9#b "6 #fx9#by9#c

#e4 xy9#bz9#c "- #cx9#byz9#b

  "6 #exy9#cz9#b

#f4 a9#b;b9#c;c9#d "- a9#c;b9#d;c9#e

  "6 #b;a9#b;b9#d;c9#d

#g4 #cy"<#bx "6 #e"> "- #dx"<#bx "6 #e">

#h4 #cy"<#d "- y">

  "6 #fx9#b"<#d "- y">

#i4 #bz9#b"<x "6 #cy">                #i




  "- #fxz"<x "6 #cy">               b#ej

#aj4 #cx"<#c "- #by">

  "- #bxy"<#c "- #by">

#aa4 #i "- a9#b

#ab4 #afx9#b "- #iy9#b

#ac4 #bbe;a9#h "- #fd;b9#b

#ad4 "<c9#f_/d9#h"> "- #aba

#ae4 x9#cy9#d "- #bexd9#f

#af4 #j4jax9#d "- #aify9#h

#ag4 "<x "6 #by">9#b "- z9#b

#ah4 "<#cx "- #by">9#b "- #bez9#b

#ai4 "<a "6 b">9#b "- "<c "6 d">9#b #bj4

#i"<#bx "- y">9#b "- #d"<#b;a "6 b">9#b

#ba4 #ha"<#dx "- #cy">9#b

  "- #be"<#cz "6 w">9#b

#bb4 x9#b "6 #fx "6 #i

  "- "<y9#b "6 #dy "6 #d">

#bc4 x9#b "- #hx "6 #af

#bd4 #d;a9#b "- #ab;ab "6 #i;b9#b

#be4 #ffxy "6 #ix9#by9#b "6 #aba

#bf4 #bx9#c "- #bhx9#b "6 #ihx

#bg4 #ez9#b "- #cjwz "6 #dew9#b

#bh4 x9<#bn> "6 #bx9ny9n "6 y9<#bn>

#bi4 "<#c "- x">9#b

  "6 #h"<#c "- x"> "6 #af            #aj




#cj4 #be "- #cj"<#bx "- #cy">       c#ej

  "6 #i"<#bx "- #cy">9#b

#ca4 a9#c "- #h

#cb4 #a "6 "<#h_/x9#i">

#cc4 #hx9<#fn> "6 #bgy9<#cm>

#cd4 x9#c "- "<y9#c_/#fd">

#ce4 #bg"<x "- y">9#c "- #h"<x "6 y">9#c

#cf4 #e"<a "- #b;b">9#c

  "- #fbe"<a "- #b;b">9#c

#cg4 x9#b "- #gx "6 #ab

#ch4 y9#b "- #by "- #h

#ci4 a9#b;b9#b "- ab "- #bj

#dj4 #bx9#b "6 #hx "6 #f

#da4 #cex9#b "- #bdx "6 #d

#db4 #cy9#b "- y "- #aj

#dc4 #f;a9#b "6 #g;a "- #bj

#dd4 #bx9#b "- #bcxy "- #ciy9#b

-------------------------------------#ea

#de4 "<x "6 y">9#b "- #g"<x "6 y">

  "6 #aj

#df4 "<y "6 z">9#b "6 "<y "6 z">

  "- #db

#dg4 #b"<#bx "6 y">9#b

  "- "<#bx "6 y"> "- #aj

#dh4 #f"<x "6 y">9#b                 #aa




  "6 #e"<x "6 y">"<y "6 z">         a#ea

  "- #f"<y "6 z">9#b

#di4 #ab"<a "6 b">9#b

  "- #ad"<a "6 b">"<c "6 d">

  "- #aj"<c "6 d">9#b

#ej4 #d"<x "- #b">9#b

  "6 #e"<x "- #b">"<y "6 #d">

  "- #ba"<y "6 #d">9#b;



  ,"! >e _m algebraic expres.ns :1 by

prop] gr\p+1 c 2 put 9to "o ( ! =ms 9 !

previ\s examples & !n factor$4



  ,,example #h4 ,factor

ax "- ay "- bx "6 by4

  .1,solu;n4 ,if1 by ! associative

axiom1 we gr\p ! f/ two t]ms tgr1 & !

la/ two tgr1 & "<use ! 4tributive

axiom"> factor \ ! common t]m1 we

trans=m ! expres.n 9to ! =m ( ,example

#b4

  ;;;ax "- ay "- bx "6 by

    _= a"<x "- y">

      "- b"<x "- y">

    _= "<x "- y">"<a "- b">4;        #ab




  ,,example #i4 ,factor             b#ea

#dx9#c "- #abx9#b "- ;x "6 #c4

  .1,solu;n4 ,ag we gr\p ! f/ two t]ms &

! la/ two t]ms4

  ;;;#dx9#c "- #abx9#b "- x "6 #c

    _= #dx9#b"<x "- #c"> "- "<x "- #c">

    _= "<x "- #c">"<#dx9#b "- #a">

    _= "<x "- #c">"<#bx "6 #a">

      "<#bx "- #a">4;



  ,9 bo? ^! examples we cd h gr\p$ ! f/

& ?ird1 & ! second & f\r? t]ms1 & obta9$

! same result4



  ,,example #aj4 ,factor #dx9#b "- #abxy

"6 #iy9#b "6 #dx "- #fy "- #c4

  .1,solu;n4 ,if we gr\p ! f/ ?ree t]ms1

! solu;n 2comes cle>4

  ;;;#dx9#b "- #abxy "6 #iy9#b "6 #dx

      "- #fy "- #c

    _= "<#bx "- #cy">9#b

      "6 #b"<#bx "- #cy"> "- #c

    _= .<"<#bx "- #cy"> "6 #c.>

      .<"<#bx "- #cy"> "- #a.>

    _= "<#bx "- #cy "6 #c">          #ac




      "<#bx "- #cy "- #a">4;        c#ea



  ,,example #aa4 ,factor

x;9#d "6 #bx9#by9#b "6 #iy9#d4

  .1,solu;n4 ,if ! coe6ici5t ( ! second

t]m 7 #f1 ! expres.n wd 2 a p]fect

squ>e4 ,"!=e1 if we add "<& subtract">

#dx9#by9#b1 \r solu;n 2comes evid5t4

  ;;;x9#d "6 #bx9#by9#b "6 #iy9#d

    _= x9#d "6 #fx9#by9#b "6 #iy9#d

      "- #dx9#by9#b

    _= "<x9#b "6 #cy9#b">9#b

      "- "<#bxy">9#b

    _= "<x9#b "6 #cy9#b "6 #bxy">

      "<x9#b "6 #cy9#b "- #bxy">4;

-------------------------------------#eb

               ,,problems



    ,factor ! foll[+ expres.ns4

;;;#a4 ax "- ay "- by "6 bx

#b4 ax "- #b;ay "- #f;by "6 #c;bx

#c4 x9#c "- #bx9#b "6 #dx "- #h

#d4 y9#c "- #by9#b "6 #ey "- #aj

#e4 #b;a "- #f "- ab9#b "6 #c;b9#b

#f4 x9#c "6 #cx9#b "- #ix "- #bg     #ad




#g4 x9#b "- #bx "6 #a "- y9#b       a#eb

#h4 xy9#c "6 #by9#b "- xy "- #b

#i4 #dx9#b "- y9#b "6 #dy "- #d

#aj4 x9#f "- #gx9#c "- #h

#aa4 x9#b "6 #bxy "6 y9#b "- z9#b

  "6 #bzw "- w9#b

#ab4 #d;a9#b "- x9#b "6 b9#b "- y9#b

  "- #dab "- #bxy

#ac4 x9#b "6 #dxy "6 #dy9#b "- x "- #by

  "- #f

#ad4 x9#c "- #ex9#b "- x "6 #e

#ae4 x9#d "- #gx9#by9#b "6 #iy9#d

#af4 y9#d "6 y9#b "6 #be

#ag4 a9#d "6 #b;a9#b;b9#b "6 #i;b9#d

#ah4 x9#d "6 #ex9#b "6 #i

#ai4 b9#d "6 #f;b9#b;c9#b "6 #be;c9#b

#bj4 #bex9#b "6 #cjxy "6 #iy9#b "6 #aex

  "6 #iy "6 #b

#ba4 #c;ax "- #f;ay "6 #d;bx "- #h;by

  "6 cx "- #b;cy

#bb4 #bjxy "6 #gzw "- #eyz "- #bhxw

#bc4 z9#d "6 #dz9#c "- #bz "- #h

#bd4 x9#d "6 #dy9#d

#be4 a9#h "- b9#h

#bf4 x9#f "6 #a                      #ae




#bg4 x9#b "6 #bxy "- z9#b "- #byz   b#eb

#bh4 "<x9#b "6 #bx "- #c">9#b "- #d

#bi4 "<x "- y "- #bz">9#b

  "- "<#bx "6 y "- z">9#b

#cj4 #b"<x "6 #b">9#b"<x "- #c">

  "6 #c"<x "6 #b">"<x "- #c">9#b;



    #c-#f ,,,simplific,n ( frac;ns,



  ,a basic pr9ciple = frac;ns1 algebraic

z well z >i?metic1 /ates t ! value ( a

frac;n is n *ang$ if xs num]ator &

denom9ator >e bo? multipli$ or bo?

divid$ by ! same quant;y "<n z]o">4

,? pr9ciple 0 /at$ 9 ,!orem #b-#h4 ,h;e1

! simplific,n or reduc;n ( a frac;n to

l[e/ t]ms is alw possible4 ,factor bo? !

num]ator & denom9ator 9to _! prime

factors &1 by us+ ! basic pr9ciple1

divide ! num]ator & denom9ator by !

product ( all _! common factors4



  ,,example #a4 ,reduce

"<#hx9#dy9#g">_/"<#abx9#fy9#c"> to l[e/

t]ms4                                #af




  .1,solu;n                         c#eb

  ;;;(#hx9#dy9#g./#abx9#fy9#c)

    _= (#b9#cx9#dy9#g

    ./#b9#b"4#cx9#fy9#c)

    _= (#b9#bx9#dy9#c"4#by9#d

    ./#b9#bx9#dy9#c"4#cx9#b)4;

,by divid+ bo? num]ator & denom9ator by

#b9#bx9#dy9#c1 we h

  ;;;(#hx9#dy9#g./#abx9#fy9#c)

    _= (#by9#d./#cx9#b)4;

-------------------------------------#ec

  ,,example #b4 ,reduce

"<x;9#b "- #gx "6 #aj">

_/"<#bx9#b "- ;x "- #f"> to l[e/ t]ms4

  .1,solu;n4 ,if we factor bo? num]ator

& denom9ator1 we h

  ;;;(x9#b "- #gx "6 #aj

    ./#bx9#b "- x "- #f)

    _= ("<x "- #e">"<x "- #b">

    ./"<#bx "6 #c">"<x "- #b">)1;

& divid+ bo? num]ator & denom9ator by

;x "- #b1 t is1 apply+ ,!orem

#b-#h1 we get

  ;;;(x9#b "- #gx "6 #aj

    ./#bx9#b "- x "- #f)             #ag




    _= (x "- #e./#bx "6 #c)4;       a#ec



  ,! elim9,n ( a common factor by divid+

! num]ator & denom9ator ( a frac;n by ?

factor is call$ .7multiplicative

c.ell,n4. ,s* a process %d 2 d"o ) c>e1

= ,!orem #b-#h is true only :5

;x "7@: #j4 ,9 ? case ! id5t;y is true =

all values ( ;x except ;x "7 #b or

;x "7 "-#c/b1 : >e n p]missible values4



  ,,example #c4 ,reduce

"<#abx9#b "6 #cjx "- #gb">

_/"<#ebx "- #hx9#b "- #fj"> to l[e/

t]ms4

  .1,solu;n4

  ;;;(#abx9#b "6 #cjx "- #gb

    ./#ebx "- #hx9#b "- #fj)

    _= (#f"<#bx "- #c">"<x "6 #d">

    ./#d"<#c "- #bx">"<x "- #e">)

    _= (#c"<x "6 #d">./#b"<#e "- x">)4;

,? id5t;y foll[s f ! fact t

#bx "- #c "7 "-"<#c "- #bx">4 "<,recall

,problem #d1 ,>ticle

#b-#d4">                             #ah




               ,,problems           b#ec



    ,reduce ! foll[+ to l[e/ t]ms4

;;;#a4 #bh/fc

#b4 (#bgx9#c./#bbex9#e)

#c4 (a9#dx9#cy./a9#bxy9#c)

#d4 (a9#b "6 ab./#c;a "6 #b;a9#c)

#e4 (a9#bx "- a9#by./ax9#b "- ay9#b)

#f4 (#bd;a9#b./#f;a9#b "- #i;a)

#g4 (x9#b "- #a./x9#b "- x)

#h4 (x9#b "- #dx "6 #d./x9#b "- #d)

#i4 (x9#b "- #af./x9#b "- #hx "6 #af)

#aj4 (a9#b "- #c;a "- #d

  ./a9#b "- a "- #ab)

#aa4 (y9#b "- y "- #f

  ./y9#b "6 #by "- #ae)

#ab4 (#bx9#b "6 #ex "- #ab

  ./#dx9#b "- #dx "- #c)

#ac4 (#f;a9#b "- #g;a "- #c

  ./#d;a9#b "- #h;a "6 #c)

#ad4 (ax "6 ay "- bx "- by

  ./am "- bm "- an "6 bn)

-------------------------------------#ed

#ae4 (#adx "- #bd "- #bx9#b

  ./x9#b "6 x "- #bj)                #ai




#af4 ("<#dx9#b "- #iy9#b">          a#ed

  "<#ahx "- #ab">

  ./"<#bx "- #cy">"<#abx "- #h">)

#ag4 (x9#b "- #cf./x9#c "- #baf)

#ah4 (#bx9#b "- #adx "6 #bj

  ./#gx "- #bx9#b "- #f)

#ai4 (#b"<x9#b "- y9#b">xy "6 x9#d

  "- y9#d

  ./x9#d "- y9#d)

#bj4 (y9#f "6 #fd

  ./y9#d "- #dy9#b "6 #af)

#ba4 (#d;a9#b "- #a

  ./#ab;a9#b "6 a "- #d;a9#c "- #c)

#bb4 (a9#b "- #b;ab "6 #c;b9#b

  ./a9#d "6 #b;a9#b;b9#b "6 #i;b9#d)

#bc ("<x9#b "- #af">

  "<x9#b "- #dx "6 #af">

  ./x9#c "6 #fd)

#bd4 (#ae;ab "- #bj;a "- #ba;b "6 #bh

  ./#ba "- a "- #aj;a9#b);



       #c-#g ,,,addi;n ( frac;ns,



  ,! algebraic sum ( two or m frac;ns

hav+ ! same denom9ator is a frac;n   #bj




) ! common denom9ator & a           b#ed

num]ator : is ! algebraic sum ( !

num]ators ( ! frac;ns 3sid]$4 ,? 0 prov$

9 ,problem #ac1 ,>ticle #b-#d4



  ,,illu/r,n4

  ;;;(#bx9#b./x "- #d) "- (#cx./x "- #d)

    "6 (#e./x "- #d)

    _= (#bx9#b "- #cx "6 #e./x "- #d)4;



  ,to f9d ! algebraic sum ( two or m

frac;ns ) di6]5t denom9ators1 we m/

replace ! frac;ns ) equival5t frac;ns

hav+ ! same denom9ators4 ,x is pref]able

to use ! .7l1/ common denom9ator.

"<,,lcd">4 ,! ,,lcd ( two or m frac;ns

3si/s ( ! product ( all ! unique prime

factors 9 ! denom9ators1 ea* ) an

expon5t equal to ! l>ge/ expon5t ) : !

factor appe>s1 & is re,y a result ( !

foll[+ important !orem4

  ,,!orem #c-#e4

  ;;;(a./b) "6 (c./d)

    _= (ad "6 bc./bd)

    "<b1 d "7@: #j">4;               #ba




  .1,pro(4 ,we h                    c#ed

  ;;;(a./b) "6 (c./d)

    _= (ad./bd) "6 (bc./bd)1;

by ,!orem #b-#h4 ,if we n[ use ,problem

#ac1 ,>ticle #b-#d1 we h

  ;;;(ad./bd) "6 (bc./bd)

    _= (ad "6 bc./bd)1;

: is \r requir$ result4



  ,,example #a4 ,f9d ! ,,lcd ( ! frac;ns

  ;;;(#cx./x9#b "- #dx "6 #d)1

    (#ex9#b./#c"<x9#b "- #d">)1

    (#b./#bx9#b "- x "- #f)4;

-------------------------------------#ee

  .1,solu;n4 ,factor+ ea* denom9ator1 we

h

  ;;;x9#b "- #dx "6 #d

    _= "<x "- #b">9#b1

  #c"<x9#b "- #d">

    _= #c"<x "6 #b">"<x "- #b">1

  #bx9#b "- x "- #f

    _= "<#bx "6 #c">"<x "- #b">4;

,! ,,lcd is #c"<x "6 #b">

"<;x "- #b">9#b"<#bx "6 #c">4

                                     #bb




  ,af ! ,,lcd has be5 det]m9$1      a#ee

equival5t frac;ns may 2 =m$4 ,divide !

,,lcd ( a giv5 frac;n by ! denom9ator (

t frac;n1 & !n multiply bo? num]ator &

denom9ator ( ! giv5 frac;n by ! result4

,! equival5t frac;ns may n[ 2 add$1 z 9

! illu/r,n abv4



  ,,example #b4 ,*ange ! foll[+ frac;ns

to equival5t "os1 ) _! ,,lcd z

denom9ator1 & f9d _! sum4

  ;;;(#d./x "6 #b)1

    (x "6 #c./x9#b "- #d)1

    (#bx "6 #a./x "- #b)4;

  .1,solu;n4 ,! ,,lcd is

"<;x "6 #b">"<x "- #b">4 ,"!=e1

  ;;;(#d./x "6 #b)

    _= (#d"<x "- #b">

    ./"<x "6 #b">"<x "- #b">)1

  (x "6 #c./x9#b "- #d)

    _= (x "6 #c

    ./"<x "6 #b">"<x "- #b">)1

  (#bx "6 #a./x "- #b)

    _= ("<#bx "6 #a">"<x "6 #b">

    ./"<x "6 #b">"<x "- #b">)1;      #bc




&                                   b#ee

  ;;;(#d./x "6 #b)

      "6 (x "6 #c./x9#b "- #d)

      "6 (#bx "6 #a./x "- #b)

    _= (#d"<x "- #b">

      ./"<x "6 #b">"<x "- #b">)

      "6 (x "6 #c

      ./"<x "6 #b">"<x "- #b">)

      "6 ("<#bx "6 #a">"<x "6 #b">

      ./"<x "6 #b">"<x "- #b">)

    _= ("<#dx "- #h"> "6 "<x "6 #c">

      "6 "<#bx9#b "6 #ex "6 #b">

      ./x9#b "- #d)

    _= (#bx9#b "6 #ajx "- #c

      ./x9#b "- #d)4;



               ,,problems



    ,reduce ! foll[+ to s+le frac;ns &

  simplify4

;;;#a4 #b/c "6 #e/f "- #c/aj

#b4 #e "- #d/i "- #g/ae

#c4 (#cx./#dy) "- (#dy./#cx)

#d4 (a9#b./b) "- (b9#b./a)

#e4 (#bx "6 #c./#f) "- (#dx "-       #bd




#g./#i) #f4 (#cx "- #a./#e)         c#ee

  "6 (#d "- #ex./#f)

-------------------------------------#ef

#g4 x "6 y "6 (x9#b./x "- y)

#h4 (x "6 #a./x "6 #b) "- (x "6 #c./x)

#i4 (#cx "- #by./#ex "- #c)

  "6 (#bx "- y./#c "- #ex)

#aj4 (#b./#abx9#b "- #c)

  "6 (#c./#bx "- #dx9#b)

#aa4 (#e./x) "- (#d./y) "6 (#c./z)

#ab4 (#d./x9#b "- #dx "- #e)

  "6 (#b./x9#b "- #a)

#ac4 (#bx "- #a./#d "- x)

  "6 (x "6 #b./#cx "- #ab)

#ad4 (x "6 #e./x9#b "6 #gx "6 #aj)

  "- (x "- #a./x9#b "6 #ex "6 #f)

#ae4 (x "- #a./#bx9#b "- #acx "6 #ae)

  "6 (x "6 #c./#bx9#b "- #aex "6 #ah)

#af4 (#bx "6 #c./#cx9#b "6 x "- #b)

  "- (#cx "- #d./#bx9#b "- #cx "- #e)

#ag4 (#c./a "- #c)

  "6 (a9#b "6 #b./a9#c "- #bg)

#ah4 (#bxy./x9#c "6 y9#c)

  "- (x./x9#b "- xy "6 y9#b)

#ai4 (#b./x9#b "6 #cx "6 #b)         #be




  "- (#c./x9#b "6 #ex "6 #f)        a#ef

  "- (#d./x9#b "6 #dx "6 #c)

#bj4 x "6 #f

  "6 (#ex "6 #a./#abx9#b "6 #ex "- #b)

  "- (x./#cx "6 #b)

#ba4 #by "- #c

  "6 (y "- #b./#dy9#b "- #aby "6 #i)

  "6 (y "6 #b./#by9#b "- y "- #c)

#bb4 (#a./"<x "- y">"<y "- z">)

  "6 (#a./"<y "- z">"<z "- x">)

  "6 (#a./"<z "- x">"<x "- y">)

#bc4 (x./"<x "- y">"<y "- z">)

  "6 (y./"<y "- z">"<z "- x">)

  "6 (z./"<z "- x">"<x "- y">)

#bd4 (#bx "- #a./#bx9#b "- x "- #f)

  "6 (x "6 #c./#fx9#b "6 x "- #ab)

  "- (#bx "- #c./#cx9#b "- #ajx "6 #h);



            ,,,5d ( sample,











                                     #bf


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