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ALGEBRA
L~~ ~~~e~~~ ~e e~~~ea~ ~a~a ~e~~e~e~~a~ ca~~idade~ c~~~cida~ ~ de-
~e~~i~ada~ .
La~ ~e~~a~ ~e e~~~ea~ ~a~a ~e~~e~e~~a~ ~~da c~a~e de ca~~idade~, ~a
~ea~ c~~~cida~ ~ de~c~~~cida~.
La~ ca~~idade~ c~~~cida~ ~e e~~~e~a~ ~~~ ~a~ ~~i~e~a~ ~e~~a~ de~ a~fa-
be~~: a, b, c, d . . .
La~ ca~~idade~ de~c~~~cida~ ~e ~e~~e~e~~a~ ~~~ ~a~ ~~~i~a~ ~e~~a~ de~
a~fabe~~ : ~, ~, ~, ~, ~, ~.
U~a ~i~~a ~e~~a ~~ede ~e~~e~e~~a~ di~~i~~~~ ~a~~~e~ dife~e~ci~~d~~~~
~~~ ~edi~ de c~~i~~a~; ~~~ e~e~~~~ : a', a", a"', ~~e ~e ~ee~ a ~~i~a, a ~e-
g~~da, a ~e~ce~a, ~ ~a~bi~~ ~~~ ~edi~ de ~~b~~dice~ ; ~~~ e~e~~~~ : a~, a2 ,
a8 , ~~e ~e ~ee~ a ~~b~~~, a ~~bd~~, a ~~b~~e~ .
OFORMULAS
C~~~ec~e~cia de ~a ge~e~a~i~aci~~ ~~e i~~~ica ~a ~e~~e~e~~aci~~ de
~a~ ca~~idade~ ~~~ ~edi~ de ~e~~a~ ~~~ ~a~ f~~~~~a~ a~geb~aica~ .
F~~~~~a a~geb~aica e~ ~a ~e~~e~e~~aci~~, ~~~ ~edi~ de ~e~~a~, de ~~a
~eg~a ~ de ~~ ~~i~ci~i~ ge~e~a~ .
A~~, ~a Ge~~e~~~a e~~e~a ~~e e~ ~~ea de ~~ ~ec~~~g~~~ e~
A = b ~ h
ig~a~ a~ ~~~d~c~~ de ~~ ba~e ~~~ ~~ a~~~~a ; ~~eg~, ~~a~a~d~ A
a~ ~~ea de~ ~ec~~~g~~~, b a ~a ba~e ~ h a ~a a~~~~a, ~a f~~~~~a/
~e~~e~e~~a~~ de ~~ ~~d~ ge~e~a~' e~ ~~ea de
c~a~~~ie~ ~ec~~~g~~~, ~~e~ e~ ~~ea de ~~ ~ec-
~~~g~~~ dad~ ~e ~b~e~d~~ c~~ ~~~~ ~~~~i~~i~
A=b~h=3 ~.X2
b ~ h e~ ~a f~~~~~a a~~e~i~~ ~~~ ~~~ ~a~~~e~
.~2 ~.=6 ~. 2 .
e~ e~ ca~~ dad~ . A~~, ~i ~a ba~e de ~~ ~ec-
~~~g~~~ e~ 3 ~ . ~ ~~ a~~~~a 2 ~ ., ~~ ~~ea ~e~~ :
E~ ~~ea de ~~~~ ~ec~~~g~~~ c~~a
A= b ~ h =8 ~4~ 34 ~. = 28 ~.2
. (1)
ba~e f~e~a 8 ~. ~ ~~ a~~~~a 31 ~ . ~e~~a : /'
OSIGNOS DEL ALGEBRA
L~~ ~ig~~~ e~~~ead~~ e~ A~geb~a ~~~ de ~~e~ c~a~e~ : Sig~~~ de O~e-
~aci~~, Sig~~~ de Re~aci~~ ~ Sig~~~ de Ag~~~aci~~ .
O6 SIGNOS DE OPERACION
E~ A~geb~a ~e ~e~ifica~ c~~ ~a~ ca~~idade~ ~a~ ~i~~a~ ~~e~aci~~e~ ~~e
e~ A~i~~~~ica : S~~a, Re~~a, M~~~i~~icaci~~, Di~i~i~~, E~~~aci~~ a P~~e~-
cia~ ~ E~~~acci~~ de Ra~ce~, ~~e ~e i~dica~ c~~ ~~~ ~ig~~~ ~ig~ie~~e~ :
E~ Sig~~ de ~a S~~a e~ +, ~~e ~e ~ee ~~~ . A~~ a + b ~e ~ee "a ~~~ b" .
(I) E~ e~ Ca~ . XVIII, ~~gi~a 270, ~e e~~~dia a~~~ia~e~~e ~~d~ ~~ ~e~aci~~ad~ c~~ ~a~
f~~~~~a~ a~geb~aica~.
~ ~PRELIMINARES
~
7
E~ Sig~~ de ~a Re~~a e~ -, ~~e ~e ~ee ~e~~~. A~~, a- b ~e ~ee "a ~e-
~~~ b"
E~ Sig~~ de ~a M~~~i~~icaci~~ e~ ~, ~~e ~e ~ee ~~~~i~~icad~ ~~~ . A~~,
a ~ b ~e ~ee "a ~~~~i~~icad~ ~~~ b" .
E~ ~~ga~ de~ ~ig~~ ~ ~~e~e e~~~ea~~e ~~ ~~~~~ e~~~e ~~~ fac~~~e~ ~
~a~bi~~ ~e i~dica ~a ~~~~i~~icaci~~ c~~~ca~d~ ~~~ fac~~~e~ e~~~e ~a~~~~e~i~ .
A~~, a .b ~ (a)(b) e~~i~a~e~ a a ~ b .
E~~~e fac~~~e~ ~i~e~a~e~ ~ e~~~e ~~ fac~~~ ~~~~~ic~ ~ ~~~ ~i~e~a~ e~
~ig~~ de ~~~~i~~icaci~~ ~~e~e ~~i~i~~e . A~~ abc e~~i~a~e a a ~ b ~ c; 5~~
e~~i~a~e a 5 ~ ~ ~ ~ .
E~ Sig~~ de ~a Di~i~i~~ e~ -, ~~e ~e ~ee di~idid~ e~~~e. A~~, a - b ~e
~ee "a di~idid~ e~~~e b". Ta~bi~~ ~e i~dica ~a di~i~i~~ ~e~a~a~d~ e~ di-
~ide~d~ ~ e~ di~i~~~ ~~~ ~~a ~a~a h~~i~~~~a~ . A~~, ~ e~~i~a~e a ~ -. ~:
0
E~ Sig~~ de ~a E~e~aci~~ a P~~e~cia e~ e~ e~~~~e~~e,
~~e e~ ~~ ~~~e~~ ~e~~e~~ c~~~cad~ a~~iba ~ a ~a de-
a3 = aaa ; b 6 = bbbbb .
~echa de ~~a ca~~idad, e~ c~a~ i~dica ~a~ ~ece~ ~~e dicha
ca~~idad, ~~a~ada ba~e, ~e ~~~a c~~~ fac~~~ . A~~,
C~a~d~ ~~a ~e~~a ~~ ~ie~e e~~~~e~~e, ~~ e~~~~e~~e e~ ~a ~~idad .
A~~, a e~~i~a~e a a~ ; ~~~ e~~i~a~e a ~'~'~' .
E~ Sig~~ de Ra~~ e~ V, ~~a~ad~ ~ig~~ ~adica~, ~ ba~~ e~~e ~ig~~ ~e c~-
~~ca ~a ca~~idad a ~a c~a~ ~e ~e e~~~ae ~a ~a~~ . A~~, -,,~a- e~~i~a~e a ~a~~ c~a-
d~ada de a, ~ ~ea, ~a ca~~idad ~~e e~e~ada a~ c~ad~ad~ ~e~~~d~ce ~a ca~-
~idad a; e~~i~a~e a ~a~~ c~bica de b, ~ ~ea ~a ca~~idad ~~e e~e~ada
a~ c~b~ ~e~~~d~ce ~a ca~~idad b .
O7 COEFICIENTE
E~ e~ ~~~d~c~~ de d~~ fac~~~e~, c~a~~~ie~a de ~~~ fac~~~e~ e~ ~~a~ad~
c~eficie~~e de~ ~~~~ fac~~~ .
A~~, e~ e~ ~~~d~c~~ 3a e~ fac~~~ 3 e~ c~eficie~~e de~ fac~~~ a e i~dica
~~e e~ fac~~~ a ~e ~~~a c~~~ ~~~a~d~ ~~e~ ~ece~, ~ ~ea 3a = a + a + a ; e~
e~ ~~~d~c~~ 5b, e~ fac~~~ 5 e~ c~eficie~~e de b e i~dica ~~e 5b=b+b-'-b+b+b .
E~~~~ ~~~ c~eficie~~e~ ~~~~~ic~~ .
E~ e~ ~~~d~c~~ ab, e~ fac~~~ a e~ c~eficie~~e de~ fac~~~ b, e i~dica ~~e
e~ fac~~~ b ~e ~~~a c~~~ ~~~a~d~ a ~ece~, ~ ~ea ab = b + b + b + b . . . a
~ece~. E~~e e~ ~~ c~eficie~~e ~i~e~a~ .
E~ e~ ~~~d~c~~ de ~~~ de d~~ fac~~~e~, ~~~ ~ ~a~i~~ de e~~~~ ~~~ e~
c~eficie~~e de ~~~ ~e~~a~~e~ . A~~, e~ e~ ~~~d~c~~ abcd, a e~ e~ c~eficie~~e
de bcd; ab e~ e~ c~eficie~~e de cd ; abc e~ e~ c~eficie~~e de d.
C~a~d~ ~~a ca~~idad ~~ ~ie~e c~eficie~~e ~~~~~ic~, ~~ c~eficie~~e
e~ ~a ~~idad . A~~, b e~~i~a~e a ~b ; abc e~~i~a~e a ~abc .
8 ~ALGEBRA
8O SIGNOS DE RELACION
Se e~~~ea~ e~~~~ ~ig~~~ ~a~a i~dica~ ~a ~e~aci~~ ~~e e~i~~e e~~~e d~~
ca~~idade~. L~~ ~~i~ci~a~e~ ~~~ :
=, ~~e ~e ~ee ig~a~ a . A~~, a = b ~e ~ee "a ig~a~ a b".
>, ~~e ~e ~ee ~a~~~ ~~e . A~~, ~ + ~ > ~ ~e ~ee "~ + ~ ~a~~~ ~~e ~".
O
<, ~~e ~e ~ee ~e~~~ ~~e . A~~, a < b + c ~e ~ee "a ~e~~~ ~~e b ~+ c" .
SIGNOS DE AGRUPACION
L~~ ~ig~~~ de ag~~~aci~~ ~~~ : e~ ~a~~~~e~i~ ~~di~a~i~ ( ), e~ ~a~~~~e-
~i~ a~g~~a~ ~ c~~che~e [ ], ~a~ ~~a~e~ ~~ ~ ~a ba~~a ~ ~~~c~~~
E~~~~ ~ig~~~ i~dica~ ~~e ~a ~~e~aci~~ c~~~cada e~~~e e~~~~ debe efec-
~~a~~e ~~i~e~~ . A~~, (a+ b)c i~dica ~~e e~ ~e~~~~ad~ de ~a ~~~a de a ~ b
debe ~~~~i~~ica~~e ~~~ c ; [a - b]~ i~dica ~~e ~a dife~e~cia e~~~e a ~ b debe
~~~~i~~ica~~e ~~~ ~ ; ~ a + b 1 _ ~ c - d ~ i~dica ~~e ~a ~~~a de a ~ b debe di-
~idi~~e e~~~e ~a dife~e~cia de c ~ d .
10 MODO DE RESOLVER LOS PROBLEMAS
EN ARITMETICA Y EN ALGEBRA
E~~~~e~~~ a c~~~i~~aci~~ ~~ e~e~~~~ ~a~a hace~ ~~~a~ ~a dife~e~cia
e~~~e e~ ~~~~d~ a~i~~~~ic~ ~ e~ a~geb~aic~ e~ ~a ~e~~~~ci~~ de ~~~b~e~a~,
f~~dad~ e~~e ~~~i~~ e~ ~a ~~~aci~~ a~geb~aica ~ e~ ~a ge~e~a~i~aci~~ ~~e
~~~a i~~~ica .
La~ edade~ de A ~ B ~~~a~ 48 a~~~. Si ~a edad de B e~ 5 ~ece~ ~a
edad de A, ~~~~ edad ~ie~e cada ~~~?
METODO ARITMETICO
Edad de A ~~~ edad de B = 48 a~~~ .
C~~~ ~a edad de B e~ 5 ~ece~ ~a de A, ~e~d~e~~~ :
Edad de A ~~~ 5 ~ece~ ~a edad de A = 48 a~~~ .
METODO ALGEBRAICO
C~~~ ~a edad de A e~ ~~a ca~~idad de~c~~~cida ~a ~e~~e~e~~~ ~~~ ~.
Sea
~ =edad de A .
E~~~~ce~ 5~ =edad de B .
C~~~ a~ba~ edade~ ~~~a~ 48 a~~~, ~e~d~e~~~ :
~ + 5~ = 48 a~~~ ;
~ ~ea,
6~ = 48 a~~~ .
O ~ea,
111 eg~,
6 ~ece~ ~a edad de A = 48 a~~~ ;
Edad de A = 8 a~~~ . R .
Edad de B = 8 a~~~ ~ 5 = 40 a~~~. R.
14 ~
ALGEBRA
Ca~~idade~ a~geb~aica~ ~~~ ~a~ ~~e e~~~e~a~ e~ ~a~~~ ab~~~~~~ de ~a~
ca~~idade~ ~ ade~~~ ~~ ~e~~id~ ~ ~a~~~ ~e~a~i~~ ~~~ ~edi~ de~ ~ig~~ .
A~~, e~c~ibie~d~ ~~e ~~a ~e~~~~a ~ie~e +$5 e~~~e~a~~~ e~ ~a~~~ ab-
~~~~~~ $5 ~ e~ ~e~~id~ ~ ~a~~~ ~e~a~i~~ (habe~) e~~~e~ad~ ~~~ e~ ~ig~~ + ;
e~c~ibie~d~ -$8 e~~~e~a~~~ e~ ~a~~~ ab~~~~~~ $8 ~ e~ ~e~~id~ ~ ~a~~~ ~e~a-
~i~~ (de~da) e~~~e~ad~ ~~~ e~ ~ig~~ - ; e~c~ibie~d~ ~~e e~ ~e~~~~e~~~ ~a~-
ca +80 ~e~e~~~ e~ ~a~~~ ab~~~~~~ 8~ ~ e~ ~a~~~ ~e~a~i~~ (~~b~e ce~~) e~~~e-
~ad~ ~~~ e~ ~ig~~ +, ~ e~c~ibie~d~ -9~ ~e~e~~~ e~ ~a~~~ ab~~~~~~ 9~ ~ e~
~a~~~ ~e~a~i~~ (ba~~ ce~~) e~~~e~ad~ ~~~ e~ ~ig~~ -.
L~~ ~ig~~~ + ~ - ~ie~e~ e~ A~geb~a d~~ a~~icaci~~e~: ~~a, i~dica~ ~a~
~~e~aci~~e~ de ~~~a ~ ~e~~a, ~ ~~~a, i~dica~ e~ ~e~~id~ ~ c~~dici~~ de ~a~
ca~~idade~ .
E~~a d~b~e a~~icaci~~ ~e di~~i~g~e ~~~~~e c~a~d~ ~~~ ~ig~~~ + ~ -
~ie~e~ ~a ~ig~ificaci~~ de ~~~a ~ ~e~~a, ~a~ e~~~e ~~~~i~~~ ~ e~~~e~i~~e~ i~-
c~~ida~ e~ ~a~~~~e~i~, c~~~ ~~~ e~e~~~~ e~ (+ 8) + (-4) ~ e~ (-7) - (+ 6) .
C~a~d~ ~a~ ~~ecedie~d~ a ~~ ~~~~i~~, ~a ~ea ~i~e~a~ ~ ~~~~~ic~, e~~~e~a~ e~
~e~~id~ ~~~i~i~~ ~ ~ega~i~~, c~~~ ~~~ e~e~~~~ e~ -a, + b, + 7, --- 8
~( REPRESENTACION GR~FICA DE LA SERIE
ALGEBRAICA DE LOS N~MEROS
Te~ie~d~ e~ c~e~~a ~~e e~ 0 e~ A~geb~a e~ ~a a~~e~cia de ~a ca~~i-
dad, ~~e ~a~ ca~~idade~ ~~~i~i~a~ ~~~ ~a~~~e~ ~~e 0 ~ ~a~ ~ega~i~a~ ~e~~-
~e~ ~~e 0, ~ ~~e ~a~ di~~a~cia~ ~edida~ hacia ~a de~echa ~ hacia a~~iba de
~~ ~~~~~ ~e c~~~ide~a~ ~~~i~i~a~ ~ hacia ~a i~~~ie~da ~ hacia aba~~ de ~~
~~~~~ ~ega~i~a~, ~a ~e~ie a~geb~aica de ~~~ ~~~e~~~ ~e ~~ede ~e~~e~e~~a~
de e~~e ~~d~ :
E~e~~~~~
-5 -4 -3-2
-1 0 +1 +2 +3 4 5
NOMENCLATURA ALGEBRAICA
17 EXPRESION ALGEBRAICA e~ ~a ~e~~e~e~~aci~~ de ~~ ~~~b~~~ a~ge-
b~aic~ ~ de ~~a ~ ~~~ ~~e~aci~~e~ a~geb~aica~ .
a, 5~, \/-4~, (a+ b )c,
(5~ - 3~)a
~2 .
~g TERMINOe~ ~~a e~~~e~i~~ a~geb~aica ~~e c~~~~a de ~~ ~~~~ ~~~b~~~
~ de ~a~i~~ ~~~b~~~~ ~~ ~e~a~ad~~ e~~~e ~~ ~~~ e~ ~ig~~ + ~ - . A~~,
a, 3b, 2~~,
4a
- ~~~ ~~~~i~~~ .
3~
NOMENCLATURA ALGEBRAICA
~
1 5
L~~ e~e~e~~~~ de ~~ ~~~~i~~ ~~~ c~a~~~ : e~ ~ig~~, e~ c~eficie~~e, ~a
~a~~e ~i~e~a~ ~ e~ g~ad~ .
P~~ e~ ~ig~~, ~~~ ~~~~i~~~ ~~~i~i~~~ ~~~ ~~e ~a~ ~~ecedid~~ de~ ~ig-
~~ + ~ ~ega~i~~~ ~~~ ~~e ~a~ ~~ecedid~~ de~ ~ig~~ - . A~~, + a, + 8~, + 9ab
~~~ ~~~~i~~~ ~~~i~i~~~ ~ - ~, - 5bc ~ -
b
~~~ ~~~~i~~~ ~ega~i~~~ .
E~ ~ig~~ + ~~e~e ~~i~i~~e de~a~~e de ~~~ ~~~~i~~~ ~~~i~i~~~ . A~~,
a e~~i~a~e a + a ; 3ab e~~i~a~e a + 3ab .
P~~ ~a~~~, c~a~d~ ~~ ~~~~i~~ ~~ ~a ~~ecedid~ de ~i~g~~ ~ig~~ e~
~~~i~i~~ .
E~ c~eficie~~e, c~~~ ~e di~~ a~~e~, e~ ~~~ c~a~~~ie~a, ge~e~a~~e~~e e~
~~i~e~~, de ~~~ fac~~~e~ de~ ~~~~i~~. A~~, e~ e~ ~~~~i~~ 5a e~ c~eficie~~e
e~ 5; e~ - 3a2~ 3' e~ c~eficie~~e e~ - 3 .
La ~a~~e ~i~e~a~ ~a c~~~~i~~~e~ ~a~ ~e~~a~ ~~e ha~a e~ e~ ~~~~i~~ . A~~,
3~ 3~4
~8~ 4
e~ 5~~ ~a ~a~~e ~i~e~a~ e~ ~~; e~ 2ab ~a ~a~~e ~i~e~a~ e~
ab .
19 EL GRADO DE UN TERMINO ~~ede ~e~ de d~~ c~a~e~ : ab~~~~~~ ~ c~~
~e~aci~~ a ~~a ~e~~a .
G~ad~ ab~~~~~~ de ~~ ~~~~i~~ e~ ~a ~~~a de ~~~ e~~~~e~~e~ de ~~~
fac~~~e~ ~i~e~a~e~ . A~~, e~ ~~~~i~~ 4a e~ de ~~i~e~ g~ad~ ~~~~~e e~ e~~~-
~ie~~e de~ fac~~~ ~i~e~a~ a e~ 1 ; e~ ~~~~i~~ ab e~ de ~eg~~d~ g~ad~ ~~~~~e
~a ~~~a de ~~~ e~~~~e~~e~ de ~~~ fac~~~e~ ~i~e~a~e~ e~ 1 + 1 = 2 ; e~ ~~~~i~~
a2b e~ de ~e~ce~ g~ad~ ~~~~~e ~a ~~~a de ~~~ e~~~~e~~e~ de ~~~ fac~~~e~
~i~e~a~e~ e~ 2 + 1 = 3; 5a 4 b 3c2 e~ de ~~~e~~ g~ad~ ~~~~~e ~a ~~~a de ~~~ e~-
~~~e~~e~ de ~~~ fac~~~e~ ~i~e~a~e~ e~ 4 + 3 + 2 = 9 .
E~ g~ad~ de ~~ ~~~~i~~ c~~ ~e~aci~~ a ~~a ~e~~a e~ e~ e~~~~e~~e de
dicha ~e~~a . A~~ e~ ~~~~i~~ b~3 e~ de ~~i~e~ g~ad~ c~~ ~e~aci~~ a b ~ de
~e~ce~ g~ad~ c~~ ~e~aci~~ a ~ ; 4~ 2~ 4 e~ de ~eg~~d~ g~ad~ c~~ ~e~aci~~ a ~
~ de c~a~~~ g~ad~ c~~ ~e~aci~~ a ~ .
20 CLASES DE TERMINOS
T~~~i~~ e~~e~~ e~ e~ ~~e ~~ ~ie~e de~~~i~ad~~ ~i~e~a~ c~~i~ 5a,
6a4b3, 2a
5
3a
T~~~i~~ f~acci~~a~i~ e~ e~ ~~e ~ie~e de~~~i~ad~~ ~i~e~a~ c~~~ b.
T~~~i~~ ~aci~~a~ e~ e~ ~~e ~~ ~ie~e ~adica~, c~~~ ~~~ e~e~~~~~ a~~e-
~i~~e~, e i~~aci~~a~ e~ ~~e ~ie~e ~adica~, c~~~ ~ab,
3b
La
"T~~~i~~~ h~~~g~~e~~ ~~~ ~~~ ~~e ~ie~e~ e~ ~i~~~ g~ad~ ab~~~~~~ .
A~~, 4~''~ ~ 6~ 2~:' ~~~ h~~~g~~e~~ ~~~~~e a~b~~ ~~~ de ~~i~~~ g~ad~
ab~~~~~~ .
T~~~i~~~ he~e~~g~~e~~ ~~~ ~~~ de di~~i~~~ g~ad~ ab~~~~~~, c~~~~ 5a,
~~e e~ de ~~i~e~ g~ad~, ~ 3a 2 , ~~e e~ de ~eg~~d~ g~ad~ .
16ALGEBRA
If EJERCICIO 4
1 . Diga~e ~~~ c~a~e de ~~~~i~~~ ~~~ ~~~ ~ig~ie~~e~ a~e~die~d~ a~ ~ig~~, a
~i ~ie~e~ ~ ~~ de~~~i~ad~~ ~ a ~i ~ie~e~ ~ ~~ ~adica~ :
2a
5b2
~
4a2b3
5a2, - 4a3b, -, -
6
. ~, -C/5b2,
6
, -
2. D~ga~e e~ g~ad~ ab~~~~~~ de ~~~ ~~~~i~~~ ~ig~ie~~e~ :
5a, -6a2b, a 2b 2, -5a3b 4 C, 8~ 5~~, 4~2~ 3 , -~~~5
3. D~ga~e e~ g~ad~ de ~~~ ~~~~i~~~ ~ig~ie~~e~ ~e~~ec~~ a cada ~~~ de ~~~
fac~~~e~ ~i~e~a~e~ :
-a3b 2 , -5~4~3, 6a 2b~3, -4abc~2 , 10~2~3b 4c5
4. De ~~~ ~~~~i~~~ ~ig~ie~~e~ e~c~ge~ c~a~~~ ~~e ~ea~ h~~~g~~e~~ ~ ~~e~
he~e~~g~~e~~ :
-4a3b 2 , 6ab 3 , -~5 , 6~ 4 ~, -2a3~4, -ab5 , 4abc~ 2, -2ac
5 . E~c~ibi~ ~~e~ ~~~~i~~~ e~~e~~~ ; d~~ f~acci~~a~i~~ ; d~~ ~~~i~i~~~, e~~e~~~ ~
~aci~~a~e~ ; ~~e~ ~ega~i~~~, f~acci~~a~i~~ e i~~aci~~a~e~ .
6. E~c~ibi~ ~~ ~~~~i~~ de cada ~~~ de ~~~ g~ad~~ ab~~~~~~~ ~ig~ie~~e~ : de
~e~ce~ g~ad~, de ~~i~~~ g~ad~, de ~~d~ci~~ g~ad~, de d~ci~~ ~~i~~~
g~ad~, de ~ig~~i~~ g~ad~ .
7. E~c~ibi~ ~~ ~~~~i~~ de d~~ fac~~~e~ ~i~e~a~e~ ~~e ~ea de c~a~~~ g~ad~ c~~
~e~aci~~ a ~a ~ ; ~~~~ de c~a~~~ fac~~~e~ ~i~e~a~e~ ~~e ~ea de ~~~~i~~
g~ad~ c~~ ~e~aci~~ a ~a ~ ; ~~~~ de ci~c~ fac~~~e~ ~i~e~a~e~ ~~e ~ea de
d~ci~~ g~ad~ c~~ ~e~aci~~ a ~a b .
CLASIFICACION DE LAS EXPRESIONES ALGEBRAICAS
21 MONOMIO e~ ~~a e~~~e~i~~ a~geb~aica
~~e c~~~~a de ~~ ~~~~ ~~~~i~~, c~~~-- -
22 POLINOMIO e~ ~~a e~~~e~i~~ a~geb~aica ~~e c~~~~a de ~~~ de ~~
~~~~i~~, c~~~~ a + b, a + ~ - ~, ~3 + 2~2 + ~ +7.
a2 5~~4
Bi~~~i~ e~ ~~ ~~~i~~~i~ ~~e
a+b, ~-~, 3
6h 2
a2
T~i~~~i~ e~ ~~ ~~~i~~~i~ ~~e
a+b+c, ~2 -5~+6, 5~2 -6~3+ 3,
c~~~~a de ~~e~ ~~~~i~~~, c~~~
23 EL GRADO de ~~ ~~~i~~~i~ ~~ede ~e~ ab~~~~~~ ~ c~~ ~e~aci~~ a ~~a
~e~~a.
G~ad~ ab~~~~~~ de ~~ ~~~i~~~i~ e~ e~ g~ad~ de ~~ ~~~~i~~ de ~a~~~
g~ad~ . A~~, e~ e~ ~~~i~~~i~ ~ 4 - 5~3 + ~ 22 - 3~ e~ ~~i~e~ ~~~~i~~ e~ de
c~a~~~ g~ad~ ; e~ ~eg~~d~, de ~e~ce~ g~ad~ ; e~ ~e~ce~~, de ~eg~~d~ g~ad~, ~
e~ ~~~i~~, de ~~i~e~ g~ad~ ; ~~eg~, e~ g~ad~ ab~~~~~~ de~ ~~~i~~~i~ e~ e~
c~a~~~ .
c~~~~a de d~~ ~~~~i~~~, c~~~ :
~
2~
3a, - 5b, 43.
a
NOMENCLATURA ALGEBRAICA~
1 7
G~ad~ de ~~ ~~~i~~~i~ c~~ ~e~aci~~ a ~~a ~e~~a e~ e~ ~a~~~ e~~~-
~e~~e de dicha ~e~~a e~ e~ ~~~i~~~i~ . A~~, e~ ~~~i~~~i~ a~ + a 4~ 2 -a2~4 e~
de ~e~~~ g~ad~ c~~ ~e~aci~~ a ~a a ~ de c~a~~~ g~ad~ c~~ ~e~aci~~ a ~a ~ .
U~ ~~~i~~~i~ e~ e~~e~~ c~a~d~ ~i~g~~~ de ~~~ ~~~~i~~~ ~ie~e de~~-
2
~i~ad~~ ~i~e~a~ c~~~ ~2 + 5~ - 6 ;
2
-
3
+
5
; f~acci~~a~i~ c~a~d~ a~g~~~
2
de ~~~ ~~~~i~~~ ~ie~e ~e~~a~ e~ e~ de~~~i~ad~~ c~~~
b
+ - 8 ; ~aci~~a~
c~a~d~ ~~ c~~~ie~e ~adica~e~, c~~~ e~ ~~~ e~e~~~~~ a~~e~i~~e~ ; i~~aci~~a~
c~a~d~ c~~~ie~e ~adica~, c~~~ V+--Ab_-~Z-VIacb ; h~~~g~~e~ c~a~d~ ~~-
d~~ ~~~ ~~~~i~~~ ~~~ de~ ~i~~~ g~ad~ ab~~~~~~, c~~~ 4a 3 +5a2b+6ab2 +b 3 ,
~ he~e~~g~~e~ c~a~d~ ~~~ ~~~~i~~~ ~~ ~~~ de~ ~i~~~ g~ad~, c~~~
~ 3 +~ 2 +~-6.
P~~i~~~i~ c~~~~e~~ c~~ ~e~aci~~ a ~~a ~e~~a e~ e~ ~~e c~~~ie~e ~~d~~
~~~ e~~~~e~~e~ ~~ce~i~~~ de dicha ~e~~a, de~de e~ ~~~ a~~~ a~ ~~~ ba~~ ~~e
~e~ga dicha ~e~~a e~ e~ ~~~i~~~i~. A~~, e~ ~~~i~~~i~ ~ 5 + ~4 - ~ 3+ ~2 - 3~
e~ c~~~~e~~ ~e~~ec~~ de ~a ~, ~~~~~e c~~~ie~e ~~d~~ ~~~ e~~~~e~~e~ ~~ce~i-
~~~ de ~a ~ de~de e~ ~~~ a~~~ 5, ha~~a e~ ~~~ ba~~ 1, ~ ~ea 5, 4, 3, 2, 1 ; e~
~~~i~~~i~ a4 - a3b + a 2b2 - ab3 + b4 e~ c~~~~e~~ ~e~~ec~~ de a ~ b .
P~~i~~~i~ ~~de~ad~ c~~ ~e~~ec~~ a ~~a ~e~~a e~ ~~ ~~~i~~~i~ e~ e~
c~a~ ~~~ e~~~~e~~e~ de ~~a ~e~~a e~c~gida, ~~a~ada ~e~~a ~~de~a~~i~, ~a~
a~~e~~a~d~ ~ di~~i~~~e~d~ .
A~~, e~ ~~~i~~~i~ ~4 - 4~3 + 2~ 2 - 5~ + 8 e~~~ ~~de~ad~ e~ ~~de~ de~-
ce~de~~e c~~ ~e~aci~~ a ~a ~e~~a ~~de~a~~i~ ~ ; e~ ~~~i~~~i~ a 5 - 2a4b + 6a3b 2
-5a2b 8 + 3ab4- b5 e~~~ ~~de~ad~ e~ ~~de~ de~ce~de~~e ~e~~ec~~ de ~a ~e~~a
~~de~a~~i~ a ~ e~ ~~de~ a~ce~de~~e ~e~~ec~~ de ~a ~e~~a ~~de~a~~i~ b.
25 O~de~a~ ~~ ~~~i~~~i~ e~ e~c~ibi~ ~~~ ~~~~i~~~ de ~~d~ ~~e ~~~ e~~~-
~e~~e~ de ~~a ~e~~a e~c~gida c~~~ ~e~~a ~~de~a~~i~ ~~ede~ e~ ~~de~ de~-
ce~de~~e ~ a~ce~de~~e . A~~, ~~de~a~ e~ ~~~i~~~i~ -5~ 8+~5-3~+~4-~ 2+6 e~
~~de~ de~ce~de~~e c~~ ~e~aci~~ a ~ ~e~~ e~c~ibi~ ~ 5 +~ 4 -5~ 3 -~ 2 -3~+6.
O~de~a~ e~ ~~~i~~~i~ ~ 4~ - 7~ 2~ 3 - 5~ 5 + 6~~4 + ~5 - ~3~2 e~ ~~de~ a~-
ce~de~~e c~~ ~e~aci~~ a ~ ~e~~ e~c~ibi~~~ :
~ 5 +6~~4--7~ 2~ 3 -~ 3~ 2 +~ 4~-5~5 .
W EJERCICIO 5
1. D~ga~e e~ g~ad~ ab~~~~~~ de ~~~ ~ig~ie~~e~ ~~~i~~~i~~ :
a) ~ 3+~ 2+~ . c) a :'b-a2b 2+ab 3-b 4 .
b) 5a-3a2+4.a 4-6-
d) ~ 5-6~4~ 3-4a2b+~2~ 4-3~~ .
2 . D~ga~e e~ g~ad~ de ~~~ ~ig~ie~~e~ ~~~i~~~i~~ c~~ ~e~aci~~ a cada ~~a
24
de ~~~ ~e~~a~ :
a) a 3+a 2-ab3 .
c) 6a 4 b 7-4a-~+ab 9-5a&b~~~ .
b) ~ 4+4~3-6~ 2~4 -4~~5 .
d) ~4~2-~~~+~~4~3-X8 +~
15-~ 11 .
CLASES DE POLINOMIOS
1 8 ~
ALGEBRA
26 T~~~i~~ i~de~e~die~~e de ~~ ~~~i~~~i~ c~~ ~e~aci~~ a ~~a ~e~~a e~
e~ ~~~~i~~ ~~e ~~ ~ie~e dicha ~e~~a .
A~~, e~ e~ ~~~i~~~i~ a3 -a2 +3a-5 e~ ~~~~i~~ i~de~e~die~~e c~~
~e~aci~~ a ~a a e~ 5 ~~~~~e ~~ ~ie~e a; e~ ~4 - 6~ 3 + 8~ 2 - 9~ + 20 e~ ~~~~i-
~~ i~de~e~die~~e e~ 20 ; e~ a3 - a2b + 3ab
2
+ b3 e~ ~~~~i~~ i~de~e~die~~e
c~~ ~e~aci~~ a ~a a e~ b 3 , ~ e~ ~~~~i~~ i~de~e~die~~e c~~ ~e~aci~~ a ~a b
e~ a3. E~ ~~~~i~~ i~de~e~die~~e c~~ ~e~aci~~ a ~~a ~e~~a ~~ede c~~~ide~a~~e
~~e ~ie~e e~a ~e~~a c~~ e~~~~e~~e ce~~, ~~~~~e c~~~ ~e ~e~~ ~~~ ade~a~~e,
~~da ca~~idad e~e~ada a ce~~ e~~i~a~e a 1 .
A~~, e~ e~ ~~i~e~ e~e~~~~ a~~e~i~~, - 5 e~~i~a~e a - 5a~, ~ e~ e~ ~~~i-
~~ e~e~~~~, b 3 e~~i~a~e a a~b 3 .
N> EJERCICIO 6
1 . A~e~die~d~ a ~i ~ie~e~ ~ ~~ de~~~i~ad~~ ~i~e~a~ ~ a ~i ~ie~e~ ~ ~~ ~adi-
ca~, d~ga~e de ~~~ c~a~e ~~~ ~~~ ~~~i~~~i~~ ~ig~ie~~e~ :
a) a 3+2a2 -3a.
c)
a ~ V~b_- 2c +
3
2
b)
a4
-~ + ~ -a.
d) da+
a
-6b+4.
2 3 2
2
2. E~c~ibi~ ~~ ~~~i~~~i~ de ~e~ce~ g~ad~ ab~~~~~~ ; de ~~i~~~ g~ad~ ab~~-
~~~~; de ~c~a~~ g~ad~ ab~~~~~~ : de deci~~~~i~~~ g~ad~ ab~~~~~~ .
3. E~c~ibi~ ~~ ~~i~~~i~ de ~eg~~d~ g~ad~ ~e~~ec~~ de ~a ~ ; ~~ ~~~i~~~i~
c) X 4~-~ 3~ 2 +~ 2~ 3-~4 .
d~ga~e c~~~e~ ~~~ c~~~~e~~~ ~ ~e~~ec~~ de c~~~e~ ~e~~a~ .
6 . E~c~ibi~ ~~e~ ~~~i~~~i~~ h~~~g~~e~~ de ~e~ce~ g~ad~ ab~~~~~~ ; c~a~~~
de ~~i~~~ g~ad~ ab~~~~~~ ; d~~ ~~~i~~~i~~ c~~~~e~~~ .
7. O~de~a~ ~~~ ~ig~ie~~e~ ~~~i~~~i~~ ~e~~ec~~ de c~a~~~ie~ ~e~~a e~ ~~de~
de~ce~de~~e :
a) ~2+6~-~3+~ 4 .
b) 6a~ 2-5a3+2a 2~+~ 3 .
c) -a 2 b 3+a
4b+a3b2-ab4 .
d) a 4 -5a+6a 3-9a 2+6 .
e) -~ 8~2+~ 10+3~ 4~~-~~~4
+~ 2~8 .
f) -3~~ 1J~ 2+4~~ 2~3-8~~~ -10~ 3~G+~ 7-7~O~ 4+~ ~~~ .
8. O~de~a~ ~~~ ~ig~ie~~e~ ~~~i~~~i~~ ~e~~ec~~ de c~a~~~ie~ ~e~~a e~ ~~de~
a~ce~de~~e :
a) a2-5a 3+6a.
d) a 2b 4+a 4b 3-a~b 2 +a."b+b 5 .
b) ~-5~ 3 +6~ 2+9~ 4 .
e) ~ 12-~~~~+~'2~ 4-~ 3~ ~~.
c) 2~'+4~ 5-O~+2~ 2+5~3 .
de ~~i~~~ g~ad~ ~e~~ec~~ de ~a a ; ~~ ~~~i~~~i~ de ~~~e~~ g~ad~ ~e~-
~ec~~ de ~a ~.
4. De ~~~
a)
~ig~ie~~e~ ~~~i~~~i~~ :
3a 2b+4a 3-5b 3 . d) 4a-5b+6c 2-8d 3-6.
b)
c)
a 4-a 3b+a2b 2 +ab 3 .
~-b~a+ab~3+ab 3~2 .
e)
f)
~ 5-a~a+a2~ 3-a 3~`-a 4~+~ 5 .
-6a3b 4 -5a~b+8a 2 b 5-b7 .
e~c~ge~ d~~ ~~e ~ea~ h~~~g~~e~~ ~ d~~ he~e~~g~~e~~ .
5. De ~~~
a)
b)
~ig~ie~~e~ ~~~i~~~i~~ :
a 4-a2+a-a3 .
5~ 4-8~ 2+~-6 .
d) ~5-~`+~0-~+5 .
e) ~5-b~'+b2~ 3-b3~ 2+b4~3
27 TERMINOS SEMEJANTES
D~~ ~ ~~~ ~~~~i~~~ ~~~ ~e~e~a~~e~ c~a~d~ ~ie~e~ ~a ~i~~a ~a~~e ~i~e-
~a~, ~ ~ea, c~a~d~ ~ie~e~ ig~a~e~ ~e~~a~ afec~ada~ de ig~a~e~ e~~~~e~~e~ .
E~e~~~~~
2a ~ a; - 2b ~ 8b; - 5a3b2 ~ - 8a8b2; ~~+1~
3~~1+i.
L~~ ~~~~i~~~ 4ab ~ - 6a2b ~~ ~~~ ~e~e~a~~e~, ~~~~~e a~~~~e ~ie~e~
ig~a~e~ ~e~~a~, ~~~a~ ~~ ~ie~e~ ~~~ ~i~~~~ e~~~~e~~e~, ~a ~~e ~a a de~ ~~i-
~e~~ ~ie~e de e~~~~e~~e 1 ~ ~a a de~ ~eg~~d~ ~ie~e de e~~~~e~~e 2 .
L~~ ~~~~i~~~ - b~4 ~ ab 4 ~~ ~~~ ~e~e~a~~e~, ~~~~~e a~~~~e ~ie~e~ ~~~
~i~~~~ e~~~~e~~e~, ~a~ ~e~~a~ ~~ ~~~ ig~a~e~ .
28 REDUCCION DE TERMINOS SEMEJANTES e~ ~~a ~~e~aci~~ ~~e ~ie-
~e ~~~ ~b~e~~ c~~~e~~i~ e~ ~~ ~~~~ ~~~~i~~ d~~ ~ ~~~ ~~~~i~~~ ~e-
~e~a~~e~ .
E~ ~a ~ed~cci~~ de ~~~~i~~~ ~e~e~a~~e~ ~~ede~ ~c~~~i~ ~~~ ~~e~ ca~~~
~ig~ie~~e~ :
1) Red~cci~~ de d~~ ~ ~~~ ~~~~i~~~ ~e~e~a~~e~ de~ ~i~~~ ~ig~~ .
REGLA
Se ~~~a~ ~~~ c~eficie~~e~, ~~~ie~d~ de~a~~e de e~~a ~~~a e~ ~i~~~
~ig~~ ~~e ~ie~e~ ~~d~~ ~ a c~~~i~~aci~~ ~e e~c~ibe ~a ~a~~e ~i~e~a~ .
E~e~~~~~
(1) 3a + 2a = 5a. R .
(6) 2ab + 3ab = eab. R .
(2) -5b-7b = -12b . R .
1
2
(7) -~~~-3~~ = -~~. R .
(3) - a2 -9a2 =-1002 . R .
(8) 5~+~+2~=8~. R .
(4) 3a~ -2 +5a~ -2=8a' -' R .
(9) - ~-3~ - 6~ - 5~ = -15~.
(5) - 4a~+ 1 -7a~'1=- 11a~+1 . R .
(10) ~~4~+~~ 2~+~~2~=?~2~. R .
2
4
8
8
. EJERCICIO 7
Red~ci~ :
1 . ~+2~ .
6 . -9~-7~ .
11-
2 . 8a+9a .
7. 4a,+5a~ .
3. 11b+9b.
8. 6a~+1+8a~
+ 1 .
12-
4 . -b-5b .
9. -~~+1-5~~+
1 .
5 . -8~-~ .
10. -3ai-2-a~-2 .
13.
REDUCCION DE TERMINO$ SEMEJANTES
1
1
2 a+ 2Q .
3ab+ 1 ab .
1~~+
8
~~ .
14.
~
19
1
4
-
5
~~ -5 ~~ .
15 . -
6
-5a2b -
8 1
a2b .
16 . -a--'~ a.
R .
2) Red~cci~~ de d~~ ~~~~i~~~ ~e~e~a~~e~ de di~~i~~~ ~ig~~ .
REGLA
Se ~e~~a~ ~~~ c~eficie~~e~, ~~~ie~d~ de~a~~e de e~~a dife~e~cia e~ ~ig~~
de~ ~a~~~ ~ a c~~~i~~aci~~ ~e e~c~ibe ~a ~a~~e ~i~e~a~ .
E~e~~~~~
(1) 2a-3a=-a. R .
(2) 18~ - 11 ~ = 7~ . R .
(5) 25a~+ 1 -54aX ` 1 = -
29a~` 1 .
1
2
1
(6) 2a - ~a = --a. R .
R .
(3) - 20ab + 11 ab = -gab .
(4) - 8aX + 13a~ = 5aX. R .
R .
(7) - 3a2b+02
b =
4a2
b. R .
~
~
R.
c
(8)- 80 X+1 +~a ~+1 =-Z ~*1
.
e
De ~a ~eg~a a~~e~i~~ ~e ded~ce ~~e d~~ ~~~~i~~~ ~e~e~a~~e~ de ig~a~e~ c~efi-
cie~~~~ ~ de ~ig~~ c~~~~a~i~ ~e a~~~a~ .
A~~ : - 8ab + B~b = 0.
2
2
5
X2~ - 5X2~=0.
R .
R .
f EJERCICIO 8
Red~ci~ :
1 . 8a-6a . 5 . 2a-2a .9.
40~3~-51~.3
~ .
2 . 6a-8a . 6 . -7b+7b .10.
-~2~+6~2~ .
3 . 9ab-15ab . 7 . -14~~+32~~ .11 .
- 15~~+40~~ .
4. 15ab-9ab .8 . -25~ 2~+3'2~ 2~ .
12. 5500-810b 2 .
20 a
ALGEBRA
17 . 8a+9a+6a . 29 . -~2~-8~2~-9~ 2~-20~2~ .
18. 15~+20~+~.30. -3a~-5a~-6a"'-9a~ .
19. -7~-8~-9~ .31.
~a
+ 9a+~a+a.
20. -a~b-a~b-3a2b .
32.~a~+1a~+
1a~ .~~a~+
21 . a~+3a~+8a~ .
22 . -5a~ + 1-3a~+ 1
-5a~ + 1 .
33 . 0.5~+0.6~+0.7~+0.8~ .
1
2 34 . - 1 ab- 1-ab---~-ab-ab .
23 . a+ 2 a+~
a . 7
14
28
2
1 35 . -
2
~3~-
1
~3~-
1
~3~-12~ 3~ .
24 . -~--~--~.
3
6
36 . ab 2+ab2+ 7ab 2+9ab2+21ab2 .
25 .6
a~+~a~+a~.37 . -~~-~-8~-77~-3~ .
38 . -~a+-1-8~a
+1-4~a + .
1-5~a+1-~a
+ 1
26 . - aa2~-6-a2~-a2~ . 1
14
639 . 1-~-Za+ a+ a+ a+ a.
27 . 11a+8a+9a+11a .
28. ~~+~+3~~"1+4~~~
+1+6,~" 1 . 40. - 1ab--'ab- ab- ab- ab .
3
(1
2
12
9
AXIOMA DE CONTINUIDAD
1 . Si ~e~e~~~ d~~ c~~~~~~~~ de ~~~e~~~ ~ea~e~ A ~ B, de ~~d~ ~~e ~~d~
~~~e~~ de A e~ ~e~~~ ~~e c~a~~~ie~ ~~~e~~ de B, e~i~~i~~ ~ie~~~e ~~ ~~~e~~
~ea~ c c~~ e~ ~~e ~e ~e~ifi~~e a :5 c :5 b, e~ ~~e a e~ ~~ ~~~e~~ ~~e e~~~
de~~~~ de~ c~~~~~~~ A, ~ b e~ ~~ ~~~e~~ ~~e e~~~ de~~~~ de~ c~~~~~~~ B .
~J!'!-1~: : ~. !C .
~, ENT\LES CON LOS NUMEROS RELATIVOS
SUMA DE NUMEROS RELATIVOS
E~ ~a ~~~a ~ adici~~ de ~~~e~~~ ~e~a~i~~~ ~~de~~~ c~~~ide~a~ c~a~~~
ca~~~: ~~~a~ d~~ ~~~e~~~ ~~~i~i~~~ ; ~~~a~ d~~ ~~~e~~~ ~ega~i~~~ ; ~~~a~ ~~
~~~i~i~~ c~~ ~~~~ ~ega~i~~, ~ ~~~a~ e~ ce~~ c~~ ~~ ~~~e~~ ~~~i~i~~ ~ ~ega~i~~ .
I)
de (~i~ ~~~ii i~, ~~ i i~~,
Reg~a
Pa~a ~~~a~ d~~ ~~~e~~~ ~~~i~i~~~ ~e ~~~cede a ~a ~~~a
(+4)+(+2)=+6
a~i~~~~ica de ~~~ ~a~~~e~ ab~~~~~~~ de a~b~~ ~~~e~~~, ~ a~
~e~~~~ad~ ~b~e~id~ ~e ~e a~~e~~~e e~ ~ig~~ + . A~~ ~e~e~~~ :
P~de~~~ ~e~~e~e~~a~ ~a ~~~a de d~~ ~~~e~~~ ~~~i~i~~~ de~ ~ig~ie~~e ~~d~ :
-4
3
'') S~~a de d~~ ~~~e~~~ ~ega~i~~~
Reg~a
Pa~a ~~~a~ d~~ ~~~e~~~ ~ega~i~~~ ~e ~~~cede a ~a ~~~a
(- 4) + (- 2) _ - 6
a~i~~~~ica de ~~~ ~a~~~e~ ab~~~~~~~ de a~b~~, ~ a~ ~e~~~~ad~
~b~e~id~ ~e ~e a~~e~~~e e~ ~ig~~ - . A~~ ~c~~ic~~~ :_ __
P~de~~~ ~e~~e~e~~a~ ~a ~~~a de d~~ ~~~e~~~ ~ega~i~~~ de~ ~ig~ie~~e
~~~~~ :
~~~~~~~ ~~~~~ . ~
-1
0 +~
NOTAS SOBRE EL CONCEPTO DE NUMERO
+4-
+Y
FIGURA 2
E- - 2
4
- 7
- 6
-S
4
-3
- 1
0
+1
2
1 3
+4
FIGURA 3
+6 ----T
+3
A
i
+4
+ 2-~
+5
i-6
+7
0 33
36ALGEBRA
Pa~a e~~~e~a~ ~a dife~e~cia (+ 4) - (- 8) = + 12, ~e~d~e~~~ :
~
-0
-8 -7 -6 -5 -4 -3 -2 -1
0 +1 +2 +3 +4
FIGURA 7
Pa~a e~~~e~a~ ~a dife~e~cia (- 8) - (+ 4) _ - 12, ~e~d~e~~~ :
MULTIPLICACION DE NUMEROS RELATIVOS
Reg~a
E~ ~~~d~c~~ de d~~ ~~~e~~~ ~e~a~i~~~ ~e ha~~a ~~~~i~~ica~d~ ~~~ ~a~~~e~
ab~~~~~~~ de a~b~~. E~ ~~~d~c~~ ha~~ad~ ~~e~a~~ ~ig~~ ~~~i~i~~ (+), ~i ~~~
~ig~~~ de a~b~~ fac~~~e~ ~~~ ig~a~e~ ; ~~e~a~~ ~ig~~ ~ega~i~~ (-), ~i ~~~ fac-
~~~e~ ~ie~e~ ~ig~~~ di~~i~~~~ . Si ~~~ de ~~~ fac~~~e~ e~ 0 e~ ~~~d~c~~ ~e~~ 0 .
+12
-12
-8 -7 -6 -5 -4 -3 -2 -1 0 +1
+2
+3
+4
C~a~d~ ~~e~a~~~ c~~ ~~~b~~~~ ~i~e~a~e~
e~ ~~~d~c~~ e~ ~ie~~~e i~dicad~, bie~ e~ ~a
f~~~a a ~ b ; bie~ e~ ~a f~~~a a . b ; ~ ~~~
~~~a~~e~~e ab .
A~~ :
i
E~ ~ig~ie~~e c~ad~~ e~ ~~ ~edi~ de ~e- + ~~~ + da + + ~~~ - da -
c~~da~ f~ci~~e~~e ~a ~e~ de ~~~ ~ig~~~ e~ ~a - ~~~ - da + - ~~~ + da -
~~~~i~~icaci~~ de ~~~ ~~~e~~~ ~e~a~i~~~ .,/'
REPRESENTACION GRAFICA DEL PRODUCTO DE DOS NUMEROS RELATIVOS
E~ ~~~d~c~~ de d~~ ~~~e~~~ ~e~a~i~~~ ~~ede e~~~e~a~~e ge~~~~~ica~e~~e
c~~~ e~ ~~ea de ~~ ~ec~~~g~~~ c~~~ ~a~g~ ~ c~~~ a~ch~ ~ie~e~ dad~~ ~~~
a~b~~ ~~~e~~~. A e~~a ~~ea ~~de~~~ a~~ib~i~~e ~~ ~a~~~ ~~~i~i~~ ~ ~ega~i~~,
(+2) (+3)=+6
(0) (+3)=0
(-2) (-3)=+6
(0) (-3)=0
(+2) (-3)=-6
00=0
(-2) (+3)=-6
THALES DE MILETO (640-535 A . C .) . E~ ~~i~e~~
~ ~~~ fa~~~~ de ~~~ ~ie~e ~abi~~ de G~ecia . S~ ~ida
e~~~ e~~~e~~a e~ ~a b~~~a de ~a ~e~e~da . F~e e~ ~~i-
~e~ fi~~~~f~ ~~~ic~ . Rec~~~i~ Egi~~~, d~~de hi~~ e~-
~~di~~, ~~~i~~d~~e e~ c~~~ac~~ de e~~e ~~d~ c~~ ~~~
SIGNOS DE AGRUPACION
45 L~~ ~ig~~~ de ag~~~aci~~ ~ ~a~~~~e~i~ ~~~ de c~a~~~ c~a~e~ : e~ ~a~~~-
~e~i~ ~~di~a~i~ ( ), e~ ~a~~~~e~i~ a~g~~a~ ~ c~~che~eL
], ~a~ ~~a~e~ ~ )
~ e~ ~~~c~~~ ~ ba~~a
46 USO DE LOS SIGNOS DE AGRUPACION
L~~ ~ig~~~ de ag~~~aci~~ ~e e~~~ea~ ~a~a i~dica~ ~~e ~a~ ca~~idade~
e~ce~~ada~ e~ e~~~~ debe~ c~~~ide~a~~e c~~~ ~~ ~~d~, ~ ~ea, c~~~ ~~a ~~~a
ca~~idad .
58
CAPITULO 111
-2~+~ c~~ ~~~ ~~~~i~~ ~ig~~~ ~ ~e~d~e~~~ :
Ve~~~, ~~e~, ~~e he~~~ ~~~~i~id~ e~ ~a~~~~e~i~ ~~ecedid~ de~ ~ig-
~~ +, de~a~d~ a cada ~~a de ~a~ ca~~idade~ ~~e e~~aba~ de~~~~ de ~~ c~~
~~ ~~~~i~ ~ig~~.
~i~~e~i~~ de ~a ~e~igi~~ egi~cia . Se ~e a~~ib~~e e~ habe~
~~edich~ e~ ec~i~~e de S~~ ~c~~~id~ e~ e~ a~~ 585 .
Ta~bi~~ ~e ~e a~~ib~~e e~ habe~ ~ea~i~ad~ ~a ~edici~~
de ~a~ ~i~~~ide~, ~edia~~e ~a~ ~~~b~a~ ~~e ~~~~ec~a~ .
F~e e~ ~~i~e~~ e~ da~ ~~a e~~~icaci~~ de ~~~ ec~i~~e~ .
A~~, a + (b - c), ~~e e~~i~a~e a a. + (+ b - c),
i~dica ~~e ~a dife~e~cia b - c debe ~~~a~~e c~~ a,
~ ~a ~abe~~~ ~~e ~a~a efec~~a~ e~~a ~~~a e~c~ibi- a + (b - c) = a + b - c.
~~~ a c~~~i~~aci~~ de a ~a~ de~~~ ca~~idade~ c~~
~~ ~~~~i~ ~ig~~ ~ ~e~d~e~~~ : ~'
La e~~~e~i~~ ~ + (-2~ + ~)
i~dica ~~e a ~ ha~ ~~e ~~~a~~e - 2~ + ~;
~ + (- 2~ +~) = ~ -2~ +~ .
~~eg~, a c~~~i~~aci~~ de ~, e~c~ibi~~~
PARENTESIS~
59
La e~~~e~i~~
a - (b + c), ~~e e~~i~a~e a a - (+ b + c),
i~dica ~~e de a ha~ ~~e ~e~~a~ ~a ~~~a b + c ~ c~~~
a - (b + c) = a - b - c.
~a~a ~e~~a~ e~c~ibi~~~ e~ ~~~~~ae~d~ c~~ ~~~ ~ig~~~ ca~-
biad~~ a c~~~i~~aci~~ de~ ~i~~e~d~, ~e~d~e~~~ :iT
La e~~~e~i~~ ~ - (- ~ + ~)
i~dica ~~e de ~ ha~ ~~e ~e~~a~ - ~ + ~; ~~eg~,
~ - (- ~ + ~) = ~ + ~ - ~ .
ca~bia~d~ ~~~ ~ig~~~ a~ ~~~~~ae~d~, ~e~d~e~~~ :
Ve~~~, ~~e~, ~~e he~~~ ~~~~i~id~ e~ ~a~~~~e~i~ ~~ecedid~ de~ ~ig-
~~ -, ca~bia~d~ e~ ~ig~~ a cada ~~a de ~a~ ca~~idade~ ~~e e~~aba~ e~ce-
~~ada~ e~ e~ ~a~~~~e~i~ .
E~ ~a~~~~e~i~ a~g~~a~ [ ~a~ ~~a~e~ ~ e~ ~~~c~~~ ~ ba~~a - -
~ie~e~ ~a ~i~~a ~ig~ificaci~~ ~~e e~ ~a~~~~e~i~ ~~di~a~i~ ~ ~e ~~~~i~e~
de~ ~i~~~ ~~d~ .
Se ~~a~ e~~~~ ~ig~~~, ~~e ~ie~e~ di~~i~~a f~~~a ~e~~ ig~a~ ~ig~ifica-
ci~~, ~a~a ~a~~~ c~a~idad e~ ~~~ ca~~~ e~ ~~e ~~a e~~~e~i~~ ~~e ~a ~ie~e
~~~ ~ ~~~ ~ig~~~ de ag~~~aci~~ ~e i~c~~~e e~ ~~~~ ~ig~~ de ag~~~aci~~ .
1 . SUPRESION DE SIGNOS DE AGRUPACION
47 REGLA GENERAL PARA SUPRIMIR SIGNOS DE AGRUPACION
1) Pa~a ~~~~i~i~ ~ig~~~ de ag~~~aci~~ ~~ecedid~~ de~ ~ig~~ + ~e de~a
e~ ~i~~~ ~ig~~ ~~e ~e~ga~ a cada ~~a de ~a~ ca~~idade~ ~~e ~e ha~~a~ de~-
~~~ de ~~ .
2) Pa~a ~~~~i~i~ ~ig~~~ de ag~~~aci~~ ~~ecedid~~ de~ ~ig~~ - ~e ca~-
bia e~ ~ig~~ a cada ~~a de ~a~ ca~~idade~ ~~e ~e ha~~a~ de~~~~ de ~~ .
E~e~~~~~
( 1) S~~~i~i~ ~~~ ~ig~~~ de ag~~~aci~~ e~ ~a e~~~e~i~~ :
a + (b - c) + 2a - (a + b) .
E~~a e~~~e~i~~ e~~i~a~e a
+a(+b-c)+2a-(+a+b).
C~~~ e~ ~~i~e~ ~a~~~~e~i~ ~a ~~ecedid~ de~ ~ig~~ + ~~ ~~~~i~i~~~ de~a~d~
a ~a~ ca~~idade~ ~~e ~e ha~~a~ de~~~~ c~~ ~~ ~~~~i~ ~ig~~ ~ c~~~ e~ ~eg~~d~
~a~~~~e~i~ ~a ~~ecidid~ de~ ~ig~~ - ~~ ~~~~i~i~~~ ca~bia~d~ e~ ~ig~~ a ~a~
ca~~idade~ ~~e ~e ha~~a~ de~~~~ ~ ~e~d~e~~~ :
a+(b-c)+2a-(a+b)=a+b-c+2a-a-b=2a-c. R .
(2) S~~~i~i~ ~~~ ~ig~~~ de ag~~~aci~~ e~ 5~ + (- ~ - ~) - [- ~ + 4~) + . ~ - 6 ~ .
E~ ~a~~~~e~i~ ~ ~a~ ~~a~e~ e~~~~ ~~e-
cedida~ de~ ~ig~~ +, ~~eg~ ~~~ ~~~~i-
~i~~~ de~a~d~ ~a~ ca~~idade~ ~~e
5~ + ( - ~ - ~) - - ~ + 4~~ + ; ~ - 6
~e ha~~a~ de~~~~ c~~ ~~ ~~~~i~ ~ig~~
= 5~ - ~ - ~ + ~ - 4~ + ~ - 6
~ c~~~ e~ c~~che~e ~a ~~ecedid~ de~
~ig~~ -, ~~ ~~~~i~i~~~ ca~bia~d~ e~
= ~- 6. R .
~ig~~ a ~a~ ca~~idade~ ~~e ~e ha~~a~
de~~~~, ~ ~e~d~e~~~ :
/
60
ALGEBRA
(3) Si~~~ifica~ ~ + 4~ - 6 + 3~ - ~ + 2~ - 1 .
E~ ~~~c~~~ ~ ba~~a e~~i~a~e a ~~ ~a~~~~e~i~ ~~e e~cie~~a a ~a~ ca~~idade~ ~~e
~e ha~~a~ deba~~ de ~~ ~ ~~ ~ig~~ e~ e~ ~ig~~ de ~a ~~i~e~a de ~a~ ca~~idade~
~~e e~~~~ deba~~ de ~~ .
fEJERCICIO 31
Si~~~ifica~, ~~~~i~ie~d~ ~~~ ~ig~~~ de ag~~~aci~~ ~ ~ed~cie~d~ ~~~~i~~~
~e~e~a~~e~ :
C~a~d~ ~~~~ ~ig~~~ de ag~~~aci~~ e~~~~ i~c~~id~~ de~~~~ de ~~~~~, c~~~ e~
e~~e e~e~~~~, ~e ~~~~i~e ~~~ e~ cada ~a~~ e~~e~a~d~ ~~~ e~ ~~~ i~~e~i~~ .
A~~, e~ e~~e ca~~, ~~~~i~i~~~ ~~i~e~~ e~ ~~~c~~~ ~ ~e~d~e~~~ :
3a+- - 5~ - [ - a + (9~ - a - ~)] ~.
S~~~i~ie~d~ e~ ~a~~~~e~i~, ~e~e~~~ : 3a + - 5~ - .-a + 9~ -
S~~~i~ie~d~ e~ c~~che~e, ~e~e~~~ : 3a + ! - 5~ + a - 9~ + a +
S~~~i~ie~d~ ~a~ ~~a~e~, ~e~e~~~ : 3a - 5~ + a - 9~ + a + ~ .
Red~cie~d~ ~~~~i~~~ ~e~e~a~~e~, ~~eda : 5a - 13~. R .
( 5) Si~~~ifica~ ~a e~~~e~i~~ :
- [-3a-~b+ [-a+(2a-b)-(-a+b)] +3b1+4a] .
E~~e~a~d~ ~~~ ~~~
~~~ i~~e~i~~e~ ~~e
~~~ ~~~ ~a~~~~e-
~i~ ~~di~a~i~~, ~e-
~e~~~ :
EJERCICIO 32
Si~~~ifica~, ~~~~i~ie~d~
~e~e~a~~e~ :
f
3a+b-a+2a-b+a-b+3b-4a
= a + 2b. R .
~~~ ~ig~~~ de ag~~~aci~~~
~ed~cie~d~ ~~~~i~~~
A~~, ~a e~~~e~i~~ a~~e~i~~ e~~i~a~e a : ~ + (4~ - 6) + 3~ - (~ + 2~ - 1) .
~+4~-6+3~-~+2~-1
S~~~i~ie~d~ ~~~ ~~~c~~~~, ~e~d~e~~~ : = ~ + 4~ - 6 + 3~ - ~ - 2~ + 1
=2~+3~-5. R .
1 . ~-(~-~). 9 . ~'2+~2-(~2+2~~+~2) +[-~'+~2] .
2 . ~2+(-3~-~2+5) . 10 . (-5~+6) +(-~+5)-6 .
3 . a+b-(-2a+3) .11 . ~+~+~-~+~-~+~-~.
4 . 4~-(-2~~-~) .
12 . a-(b+a) +( -a+b)- (-a+2b) .
13 . -(~--~2)+~~ +(-2~2+3~~)- [-~2+~~] .5 . 2~+3~-4~+3~ .
6 . a+(a-b)+(-a+b) . 14 . 8~ 2+[-2~~+~2 ]-] -~2 +~~-3~ 2 ~-(~ 2 -3~~)
7 . a 2+[-b2+2a2]-[a 2- b 2 ] . 15 . -(a+b)+(-a-b)- ( -b+a)+(3a+b) .
8 . 2a-~ -~+a-1 -~ a+~-3 ~ .
(4) Si~~~ifica~ ~a e~~~e~i~~ : 3a + ~ - 5~ - [ - a + (9~ - a + ~)] ~ .
1 . 2a+[a-(a+b)] . 4 . 4~2+[-(~2-~~)+( -3~2+2~~)- (-3~2+~2)) .
2 . 3~-[~+~-2~+~] .5 . a+i ( -2a+b)-(-a+b-c)+a ~.
3 . 2~-[(~-~)-(~+~)] . 6 . 4~-[2~+~-3]+[-4~-2~+1] .
- -3a - ;b +' -a+2a-b+a-b1+3b ~: +4a)
_- -3a- ; b-a+2a-b+a-b +3b +4a]
_- _ - 3a -b+~-2a+b-a+b-3b+4a'
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