,,,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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