ALGEBRA (6to grado

ALGEBRA - JUNIO JUNIO JULIOAGOSTO-AGOSTO SACO OLIVEROS PRIMARIA LOCUTORIO REN@TRIX CEL :992444616

INSTITUCIÓN EDUCATIVA PRIVADA CRISTIANA “CRISTO REDENTOR “

“ S i n o p u e d e s te n e r aq u e llo q u e h u b ie r a s a p r e c ia d o , a p r e c i a a q u e l lo lo q u e t i e n e s ”

ALGEBRA M E S D E :


ÁLGEBRA


Niels Heni! A"el# n$%i&' el ( &e A)'s*' &e 1+,2# - ./e*' el 16 &e A"il &e 1+29# 0/e /n "ill$n*e .$*e.*i%' n'/e)' G$n3 /n $n%' e%'n'%i.ien*' $ l$ e&$& &e 1+ %'n s/ 5i.e *$"$'# en 7/e 5'"3 7/e l$ e%/$%i3n )ene$l &e 7/in*' )$&' es ins'l/"le 5' 5'%e&i.ien*'s $l)e"$i%'s Abel fue ins!u"en#l en es#ble$e! el #n%lisis "#e"%i$&s en un# b#se !i'u!&s#( En su "#)&! !#b#*& +Re$,e!$,es f&n$i&ns ellii.ues+ /In0esi'#$i&nes en fun$i&nes el1i$#s2 345672 !e0&lu$i&n8 l# $&"!ensi8n 9e l#s fun$i&nes el1i$#s #l esu9i#! el uni0e!s& 9e es#s fun$i&nes(

EPRESIÓN ALGEBRAICA


Es /n %'n/n*' &e %'ns*$n*es - 8$i$"les %'n e5'nen*es $%i'n$les# el$%i'n$&'s $ *$8s &e l$s '5e$%i'nes &e s/.$# es*$# ./l*i5li%$%i3n# &i8isi3n# 5'*en%i$%i3n $&i%$%i3n# es /n n;.e'
4 x 2 = 2 x > 1 2 x > 2 xy = >2 >2 xy = x y =

1

ELEMENTOS DE UNA EPRESIÓN ALGEBRAICA

2

4 x y C ' e < % ie n * e



E5'nen*es

V$i$"les

CLASI;ICACIÓN MONOMIO:

Un .'n'.i' &e /n$ 8$i$"le es /n$ e5esi3n &e l$ 0'.$: n

ax

?'n&e $ es /n$ %'ns'n$n*e %'e<%ien*e &el .'n'.i' - n es /n en*e' 5'si*i8' POLINOMIO: Es /n$ e5esi3n $l)e"$i%$ 7/e %'ns*$ &e &'s ' .s *.in's en /n$ %$n*i&$&
A l's 5'lin'.i's &e &'s *.in's se les &en'.in$ BINOMIOS# $ l's &e *es *.in's TRINOMIOS $ l's &e %/$*' *.in's CUATRINOMIOS en )ene$l se les ll$.$ POLINOMIOS 2  → Bin'.i' ( x + 6x E*e"l&s :

 + x − x + 6 → Tin'.i'  P'lin'.i'  4 2 D x − x + 6x − 4 → C/$*in'.i'   

2

I( GRADOS DE UN MONOMIO 3( GRADO ABSOLUTO DE UN MONOMIO /G(A(7


Es* &$&' 5' l$ s/.$ &e e5'nen*es &e s/s 8$i$"les 5( GRADO RELATIVO DE UN MONOMIO /G(R(7 Es* &$&' 5' el e5'nen*e &e l$ 8$i$"le e0ei&$ E*e"l& :

GA = 2 + ( + 1 = +

( 2 ( x y z D

GR x  = 2 GR y =( GR z   =1

II( GRADOS DE UN POLINOMIO /G(A(7 3( GRADO ABSOLUTO DE UN POLINOMIO /G(A(7 Es* &$&' 5' el MAOR GRA?O &e l's .'n'.i's 5( GRADO RELATIVO DE UN POLINOMIO /G(R(7 Es* &$&' 5' el MAOR ?E LOS EXPONENTES &e l$ 8$i$"le e0ei&$ E*e"l&s : A7  4

( x y

?$&' el P'lin'.i' : − Dx 2y4 + 2x6y2

↓ ↓ GA = D GA = 6

B7

GA = +

− 1x4y

↓

↓

GA = +

GA = (

( x y z

↓ GA = 11

2 4 

− 6x y z

GR@ y A = 4 G A = 14

?$&' el P'lin'.i' : 4 ( 2

GR@ x A = 6

D 2 (

+ x y z

↓

↓

GA = 9

GA = 14

GR@ x  = D GR@ y  = ( GR@ z  = (

<#ll#: 3( El )$&' el$*i8' &e %$&$ 5'liln'.i' %'n es5e%*' $ l$ 8$i$"le F x  5( El )$&' $"s'l/*' &e %$&$ 5'lin'.i'


G . R .(x 

P & lin & " i& ( x 6 > D x 4 > C x 2 = 6 1 1 x C > 6 x 4 = ( x 6 > + 2 x y ( > + x 2 y 4 > ( x C y 6 > D C x C y 2 z > x 4 y C z 2 > 1 6 2 x $ y b > ( x 2 $ = 1 y b > 2 = 6 x 2 $ = 1 y b = 1 , # 4 x

$= C

y " = 2 > , # 6 x

$= ( " = +

y

= x $ = 1 y "

+ x y z 9 > 9 x + y z C > 6 x y 4 z 6 > 1 1 6 a x 2 y = 9 a x y C > ( x C y

2

+ x C y z > ( x 4 y z 6 = x 2 y C z ( C b x C y

4

– D b C x 2 y C – 4 x 4 y C

C ( C 2 1 C 2 x y b + x y z – x 4 y 2 z 6 ( C C x 9 y 1 , = 2 C

x 5 y 6

=

2 x ( y

11

= 6 x 1 , y D

6 x 8 y 4 – 4 x 2 y 3

m x 2 y C – 2 n x y 4

= 6 x ( y 2

x ( y 2 z + x 1 , y 1 2 – D x 2 y 3 z (

G (A (


<#ll# : 3( El )$&' el$*i8' &e %$&$ 5'lin'.i' %'n es5e%*' $ l$ 8$i$"le . 5( El )$&' $"s'l/*' &e %$&$ 5'lin'.i'

P & lin & " i& 6

+ m 4n

2

G ( R ( /" 7

– m 2n

4

+ bm



#7

m

b7

b 4m + b (m

$7

( a 2b m + a b 4m

97

x 4 m = 4 m

e7

D m 4n

f 7

9 a 4b Dm

'7

4 a  x m

,7

4 m 2 n  x – 2 m 4 n ( x y = m 6 n

i 7

m

* 7

9m

=7

x y

l 7

9 x 6 y

" 7

m 4n

n7 &7

D

2

9



1,

(

= m

2

+ b

D

6

4

+ m D n 4 y

– 4 a ( x = a b 9 x 1 , D

= m

+

D

D

9

– m D n 4 y D

– m n

6

(

= 2 x D y

= m x 4 y

6 a 4b Dm = ab 9m D

+

= x n

= m 2 x 2 y

1 4 a  b c 6 x

4

= ab

= m n

– 6m 2

(

– 6 x 2 y

– m n

= m 9

2

(

6



– ( a b

– 11m x

(

=

+

( x ( y

4

G(A(


T>RMINOS SEMEJANTES S'n $7/ell's 7/e *ienen l$ .is.$ 5$*e 8$i$"le 2 

T1

E*e"l& :

2 

4

D x y

?

( xy

x y

T2

 T 

T 1

T C

4 4

2 

2 x y T4

T (

+ x y



T(

x 2y 

#7

L's *.in's # l' *$n*' s'n SEMEJANTES(

b7

L's *.in's T 2 - T 4 n' *ienen l$ .is.$ 5$*e 8$i$"le# 5' l' *$n*' NO SON SEMEJANTES(

*ienen l$ .is.$ 5$*e 8$i$"le

# 5'

REDUCCIÓN DE T>RMINOS SEMEJANTES :

Es /n 5'%es' 7/e %'nsis*e en *$ns0'.$ &'s ' .s *.in's se.e$n*es en /n' s'l'# s/.$n&' ' es*$n&' l's %'e<%ien*es - es%i"ien&' $ %'n*in/$%i3n &el es/l*$&' l$ .is.$ PARTE VARIABLE 7/e $5$e%e en l's *.in's E*e"l&s :

Re&/%i : 3(

+ x

2

+

6x

2

−

Cx

2

=

@+ + 6 − CAx

2

=

2

11x

1  m = 6 n > D m > 4 n J @1  > D A m = @6 > 4 A n J 6 m = 2 n

5( D x 2 y D = 1 , x y D = 2 x y D > 4 x 2 y D J @ D > 4 A x 2 y D = @ 1 , = 2 A x y D J  x 2 y D= 1 2 x y D

@(


Re9u$e l&s si'uienes !"in&s se"e*#nes : 3(

2 2 2 2 x y > x y =  x y

J

5(

1+ x > 1, x > =D x

J

@(

14m > m = 4m

J

(

n y =

(

4 x > 21 x

(

+a = 9a > 16a

6(

ab

4(

m

(

 x = ( x > 6 x

3F(

  2 26 x > + x = 9 x

33(

2  4  4 2 D x y = + x y > ( x y > 6 x y J

35(

6 ( 6 (m = 4m > 2m

3@(

D ( 4 D ( 4 m n = 2m n > m n > m n J

3(

D 2 D 2 ( ( (a b > a b = 1,ax > (ax J

3(

4 4 2 4 4 2 + x y = D x y > 6 x y > 2 x y

2

2 2 2,n y > 19n y

J J J

2 2 2 = Dab = 9ab

J

( ( 4 4 = 4m = 6m > 2m J m

m

m

J J

J

( 6 ( 3( x = 4 x = 2 x

J J

36(

( 4 4 . = . = .

J

34(

2  4 x =  x = ( x = 6 x

J

3(

+ 4 9 + + >4 Dm n > (m n = 2m n

J

5F(

( 6 ( 4 xy = 2 xy > xy

J


“ ! n ic a " e n te la o b e d ie n c i a, tie n e d e re c h o a l " a n d o”

ALGEBRA M ES D E :


ÁLGEBRA LAGRA#GE, $%SE&' L%!(S )E


El 0Ksi%' 0$n%s 'se5 L'/is# %'n&e &e L$)$n)e# n$%i&' en Ene 2(# 1D6# ./e*' en A" 1,# 1+1# 0/e /n' &e l's %ien*K<%'s .$*e.*i%'s - 0Ksi%'s .s i.5'*$n*es &e
OPERACIONES CON POLINOMIOS 37 ADICIÓN DE POLINOMIOS


 L$ s/.$ s/ele in&i%$se in%l/-en&' l's s/.$n&'s &en*' &el 5$n*esis $sK: E*e"l&:

?$&' l's 5'lin'.i's: A = x B= x

(

2

2

A + B = x

+

xy + (y 2

−

2xy − 2y2

2

+

xy + (y2 )

+

H$ll$ A=B

( x2

−

2xy − 2y 2

)

A'$ %'l'%$.'s *'&'s l's *.in's &e es*'s 5'lin'.i's /n's $ %'n*in/$%i3n &e '*'s %'n s/s 5'5i's si)n's - *en&e.'s: 2 2 A + B =  x +  x y + ( y + x 2 – 2 x y – 2 y

2

A=B J 42 = - = -2 

En l$ 5%*i%$# s/elen %'l'%$se l's 5'lin'.i's /n's &e"$' &e '*'s &e .'&' 7/e l's *.in's se.e$n*es 7/e&en en %'l/.n$# l/e)' se e0e%*;$ l$ e&/%%i3n &e &i%'s *.in's 2

2

A = Cx + Cxy + (y 2 2 B = x − 2xy − 2y A + B = 4x

2

+

2

xy + Cy

OBSERVACIÓN: Un si)n' &e $)/5$%i3n 5e%e&i&' &el si)n' = se eli.in$# sin %$."i$ &e si)n' $ *'&'s l's *.in's es%i*'s &en*' &el si)n' &e $)/5$%i3n

57 SUSTRACCIÓN DE POLINOMIOS E*e"l&:

?$&'s 5'lin'.i's P J 64 = 42 = 4


J> 44 = 22 =  

H$ll$ P > 

L$ s/s*$%%i3n se in&i%$ in%l/-en&' el s/s*$en&' en /n 5$n*esis 5e%e&i&' &el si)n'  $sK: P > J 64 = 42 = 4 > > 4 4 = 22 =  A'$ &e$.'s el .in/en&' %'n s/ 5'5i' si)n' - $ %'n*in/$%i3n es%i"i.'s el s/s*$en&' %$."in&'le el si)n' $ *'&'s s/s *.in's P – Q = 6 x4+ 4 x2+ 4 + 4 x

4

– 2 x

2

– 

P >  J 1,4 = 22 = 1 

En l$ 5%*i%$# s/ele es%i"ise el s/s*$en&' %'n s/s si)n's %$."i$&'s &e"$' &el .in/en&'# &e .'&' 7/e l's *.in's se.e$n*es 7/e&en en %'l/.n$# l/e)' se e0%*;$ l$ e&/%%i3n &e &i%'s *.in's 4

2

P = 6x + 4x 2 + 4 4 B = 4x − 2x −  P − Q = 1,x 4 + 2x 2 + 1

OBSERVACIÓN: Un si)n' &e $)/5$%i3n 5e%e&i&' &el si)n' > se eli.in$# %$."i$n&' &e si)n' $ *'&'s l's *.in's es%i*'s &en*' &el si)n' &e $)/5$%i3n IMPORTANTE

(+) (+ ) = + (−) (− ) = + (+) (− ) = − (−) (+ ) = −

E*e"l& : E0e%*/$ : + x 2 = D x = 6 >>2 x 2 = ( x > 9 S&lu$i8n : 2

+ x + D x + 6 + 2 x

2

– ( x + 9


⇒

2

1, x + 2x + 1(

3(

?$&'s l's 5'lin'.i's :  2 A J ( x = 6 x = 6 x = 9  2 B J >2 x > 2 x > 4 x = 6  2 C J x >  x =  x > + H$ll$ A = B = C

5(

?$&'s l's 5'lin'.i's : 4 2  A J  x = + x = 2 x = x = 6 2  4 B J 6 x > x = + = ( x 4 2 C J 9 x > D x = 1 x > 4 C$l%/l$ A = B = C

@(

H$ll$ A > B s$"ien&' 7/e :  2 A J 4 x = ( x = x = + 2 B J > x = 6

(

H$ll$ A > B s$"ien&' 7/e : 2 4 A J 1, x > D x = 6 x = 9 2 B J 4 x > ( =  x


(

?$&'s l's 5'lin'.i's : ( 4 A J  x = 2 x = 6 x = 16 B J

1, x 4 = 2 x  > ( x = 4

( 4  C J 2 x > + x = x =

12

C$l%/l$ :  A + B – C

(

3(

Eli.in$ l's si)n's &e $)/5$%i3n - $ll$ el es/l*$&' : $

6 x 4 >  x 4 > 2 x = 1 J

"

2 x  > > 4 x > 2 x  J

%

D x 4 >  6 x > ( > 2 x 4 J

&

+ x  >  x 4 = 1 = 2 x 2 =  x 2 = ( J

e

( x  > 2 x  > 4 =  x 2 = 6 J

0

 x 4 > >  x 4 = 6 x 2 = x > 2 x 4 =  J

)

+ x 4 = > ( x 4 > 2 x 4 >  x = 4 J

?$&'s l's 5'lin'.i's : A J  x ( = 2 x 4 = D x 2 = + x = 9


B J ( x 4 = + x  > ( x 2 >  x = 4 C J 2 x ( > 2 x 2 > 6 H$ll$ : $ A + B " A + C % A + B + C 5(

?$&'s l's 5'lin'.i's : 4 2  A J  x = 2 x = 6 x = + 2 B J D x = 9 x = 11  C J >D x = ( x 2 4 D J x > 4 x = 1 H$ll$ : $ A + B " B + C & B – D e A + B + C

@(

& A – C

% B + D 0 A – C

Eli.in$ l's si)n's &e $)/5$%i3n - $ll$ el es/l*$&' : $

a + b = b + c = c + d = a – c J

"

( x = D y = + > 2 x =  y > 4

J

%

a + b + c  = 2a = 2b > c

J

&

m2 = 2mn > mn = n2

J

e

 x  = + xy 2 = y  > ( xy 2 = x  > y 

J

0

(ab > bc = 4cd = 2ab > cd

J

MULTIPLICACIÓN DE EPRESIONES ALGEBRAICAS P$$ ./l*i5li%$ e5esi'nes $l)e"$i%$s# es*/&i$e.'s l's si)/ien*es %$s's : MULTIPLICACIÓN DE MONOMIOS Se ./l*i5li%$n l's %'e<%ien*es - l$s 5$*es li*e$les &e %$&$ .'n'.i' E*e"( :

M/l*i5li%$ :

2a2 a 2a2 a J 2a2a J 6a2= J 6a(


MULTIPLICACIÓN DE UN MONOMIO POR UN POLINOMIO Se ./l*i5li%$ el .'n'.i' 5' %$&$ /n' &e l's *.in's &el 5'lin'.i'# *enien&' en %/en*$# en %$&$ %$s'# l$ e)l$ &e l's si)n's - se se5$$n l's 5'&/%*'s 5$%i$les %'n s/s 5'5i's si)n's E*e"( :

M/l*i5li%$ :  x 2 > 6 x = D 4ax 2  x 2 > 6 x = D 4ax 2

J  x 2 4ax 2 > 6 x 4ax 2 = D 4ax 2 J 12ax 4 > 24ax  = 2+ax 2

MULTIPLICACIÓN DE UN BINOMIO POR UN POLINOMIO Se ./l*i5li%$n *'&'s l's *.in's &el 1e 0$%*' 5' %$&$ /n' &e l's *.in's &el 2&' 0$%*' - SE RE?UCEN LOS TWRMINOS SEMEANTES E*e"( :

M/l*i5li%$ : $2  2$ = $ $ = 1 Eis*en 2 0'.$s &e &es$'ll$ 7/e s'n : &ri"era *or"a+ a2 > 2a = a a = 1 J 2  2     a a  = a 1 > 2a a  > 2a 1 = a a  = a 1 J (  4   a = a > 2a > 2a = a = a O&en$n&' : ( 2 4 6  a = a > 2a > 2a = a = a

Seunda *or"a+

O&en$n&' :

6 ( 4 a = a > 2a

a2 > 2a = a a = 1 J

 2 a = a > 2a  a = 1

 = a = a2 > 2a




6 ( 4  2 a = a > 2a = a = a > 2a

M/l*i5li%$: 3(

 x 2= xy = y 2  x > y 

@(

a2 = b2 = 2ab a = b

(

 x  >  x 2 = 1  x = 

(

a > a = a2 a > 1

(

m4 = m2n2 = n4 m2 >

n

5(

a2=b2  2ab a – b

2



6(

 x  > 2 x 2 =  x > 1 2 x = 

3(

m > m2 = m am = a

4(

 y  = ( > 6 y   y 2 = 2


5(

a2 > (ab = 2b2 4a > (b

@(

(m4 > m2n2 = n4 m > n

(

a2 = a = 1 a = 1

(

 x  = 2 x 2 > x   x 2 > 2 x 

(

 x 2 = 1 = x   x 2 = x 

6(

m > 4m = m2 m = 1

4(

n2 > 2n = 1 n2 > 1

(

2 y  = y >  y 2 2 y = (

3F(  x  > a = 2$2 2a2 > x 2

LÓ GICO M AT EM Á TICO

6 * '  G  $& '

ALGEBRA



GAUSS2 CARL C$l Yie&i% G$/ss# n$%i&' el , &e A"il &e 1DDD ./e*' el 2 &e Ye"e' &e 1+((# 0/e /n .$*e.*i%' $le.n 7/ien &'.in3 l$ %'./ni&$& .$*e.*i%$ &/$n*e - &es5/s &e s/ 8i&$ Un 5'&i)i' &e niQ'# G$/ss $5en&i3 $ lee - l$ $i*.*i%$ $ l$ e&$& &e *es $Q's Al e%'n'%e s/ *$len*'# el ?/7/e &e B/nsZi%! en 1D92 le 5'8e-3 %'n /n s/el&' 5$$ $sK 5e.i*ile se)/i s/ e&/%$%i3n Mien*$s *'&$8K$ $sis*e $ l$


Uni8esi&$& &e C$'line 1D929(# G$/ss 0'./l3 el .*'&' &e l's .en's %/$&$&'s - /n$ %'ne*/$ en l$ &is*i"/%i3n &e n;.e's 5i.'s en*e *'&'s l's n;.e's el .s e%ien*e 0/e 5'"$&' 5' $%7/es H$&$.$& en 1+96 ?/$n*e es*e 5eK'&' G$/ss n' *enK$ $%%es' $ /n$ "/en$ "i"li'*e%$ .$*e.*i%$ - 5' es' e&es%/"i3 ./%'s *e'e.$s -$ $%e5*$&'s L$ si*/$%i3n %$."i3 en 1D9(# %/$n&' 0/e $ G'**in)en %'n s/ e%elen*e "i"li'*e%$ En 1D9( G$/ss &es%/"i3 el *e'e.$ 0/n&$.en*$l &e esi&/'s %/$&*i%'s# 7/e *$*$n &el %'n%e5*' &e %'n)/en%i$ en l$ *e'K$ &el n;.e' En 1D96 i' s/ 5i.e$ .$%$ %'.' /n .$*e.*i%' sei' 5' 5'"$ l$ 5'si"ili&$& &e %'ns*/i /n 5'lK)'n' e)/l$ &e 1D l$&'s /s$n&' s3l' /n$ e)l$ - /n %'.5s L's 53i.'s 4 $Q's le 0/e'n ./- 5'&/%*i8's Le 8enK$n i&e$s *$n 5i&$.en*e 7/e 5'&K$ se)/i s3l' $l)/n$s &e ell$s En 1D99 l$ Uni8esi&$& &e Hel.s*e&* le %'n%e&i3 $ G$/ss /n P? )$&' 5' /n$ &ise*$%i3n 7/e &i' l$ 5i.e$ 5/e"$ &el *e'e.$ 0/n&$.en*$l &el l)e"$ G$/ss */8' &'s e$li$%i'nes .$-'es en 1+,1 L$ 5i.e$ 0/e l$ 5/"li%$%i3n &e s/ ?is7/isi%i'nes $i*.*i%$s# /n *$*$&' en *e'K$ &el n;.e'# 7/e %'n*/8' s/s s'l/%i'nes $ ./%'s 5'"le.$s sin li7/i&$ Es*e li"' <$ "$ses 5$$ in8es*i)$%i'nes 0/*/$s &n&'le /n .$-' e%'n'%i.ien*' en*e l's .$*e.*i%'s &e s/ *ie.5' L$ se)/n&$ 0/e &e"i&' $l &es%/"i.ien*' &el $s*e'i&e Cees Se $"K$ '"se8$&' "e8e.en*e en el ene' &e 1+,1 5e' en*'n%es $"K$ &es$5$e%i&' &e 8is*$ G$/ss %$l%/l3 l$ 3"i*$ /s$n&' /n$ .e' *e'K$ - 5e&i' &'n&e - %/$n&' Cees e$5$e%eK$ C/$n&' l$ 5e&i%%i3n 0/e 5'"$&$ %'e%*$# l$ 0$.$ &e G$/ss se e*en&i3 $ l' le's - $n%' S/"si)/ien*e.en*e $%e5*3 /n$ se)/$ 5'si%i3n
S#bi#s .ue

L$s e$s &e l$s <)/$s )e'.*i%$s n's 5e.i*en &e.'s*$ i&en*i&$&es $l)e"$i%$s Ee.5l':


a–b a a b –

=

a–b

a

a $2 − "2 = @$ + "A@$ − "A

DIVISIÓN DE MONOMIOS P$$ &i8i&i .'n'.i's# 5'%e&e.'s $ &i8i&i l's %'e<%ien*es - $5li%$.'s l$ *e'K$ &e e5'nen*es &i8isi3n &e "$ses i)/$les 5$$ l$ 5$*e li*e$l

Ee.5l':

6( x(y+ z 1, ( xy2z 1,

1 x2y6z,

=

1x2y 6

=

DIVISIÓN DE UN POLINOMIO ENTRE UN MONOMIO El 5'%e&i.ien*' 5$$ &i8i&i /n 5'lin'.i' en*e /n .'n'.i' es el .is.' 7/e e$li$.'s en l$ &i8isi3n en*e .'n'.i's s3l' 7/e $'$ el .'n'.i' &i8i&i $ %$&$ *.in' &el 5'lin'.i' Ee.5l': C, x(y9 + 1,x4yD − 4 x1,y1( 2 xy(

=

=

16 x4 + 24x( − 2x −+ x

16x4  − +x = −2 x

=

= −

2 x

24x(  − +x 2 − x

+

−

x 2

C,x(y9 2xy(

+

1,x4 yD 2xy(

4 x1,y 1( 2xy (

1( x2y4 + (x1y2 − 2xDy 1,

2x   − +x , + 4 x

−

+

−

4

PRACTIHUEMOS

L e - & e s i) n ' s : N ' *$ : @ > A @ > AJ @= A @= A@= AJ @= A @ > A@= AJ @ > A @ = A @ > AJ @ > A


?i8i&i : 1

a2 − ab a



a > (ab2 > 6a2b >a

2

4 x+ > 1,x6 > 6x 4

D

2a4b+

16 x6y12 > 2,x1,y1+ + C2x+ y 2,

+

12 1+

1, 16

24 x y > 1+x y

+

12 1D

1,x y

D 12

2 x y

+

1, + C 4 1(x y > (x y C 4 ( x y

9

>4 x4 y 1,

6

( x6y4 > 1,x(y12 + (x1,y 1 ( x4 y 

( 9 C, x y

2 x 

4

2,a1,b1( > 1+a16b1, + 14a+b12

(

9 x2y − 6a2y 4 x 2 −

4 x6 yD > 2x y1, + + x(y 4 2 x2y

1,

TRABAJEMOS EN CASA ?i8i&i : 44 x4 y12 > 24 x9y1, + 16x6y D

12 xDy( > 6x6yD + 1(xD y 6

1

 x2y

2

>4 xy 6


1, x6 yD > (x12y1+ + 1(x1+ y12 > 2,xDy 9 ( x2y (



2 x( y6 > + x6 y1, + 4 x4 y (

12 x( y6 > 4+x9y1( + 6xDy 1,

1, x6y( > 1(x4 y6 + (x y ( ( xy 

(

9

6 x4 y (

6

2, x4 y( > 12xy9 + 16xD y 1, 4 xy 4

D

2 x y 4

4

2 x1,y12 > 64 x2(y1D + 16x12y 1, + xD y +

+

16 x+y D 2 x(y 2

1,

9 x(y D xy 6 −

PRODUCTOS NOTABLES S'n $7/ell's 5'&/%*'s 7/e $l 5esen*$ %ie*$ 0'.$ 5$*i%/l$# e8i*$ 7/e se e0e%*/ l$ '5e$%i3n &e ./l*i5li%$%i3n es%i"ien&' &ie%*$.en*e el es/l*$&'

I(

CUADRADO DE UN BINOMIO


( a + b)

2

( a − b)

2

=

a2 + 2ab + b2

=

a2 − 2ab + b2

Ee.5l's: ( x + ()

2

( y − C)

2

( x)

=

( y)

1

y2

2

( Z + )

2

2

2

2( x ) ( () + ( () x2 + 1,x + 2(

=

2

+

−

2( y ) ( C)

−

6y

2

=

4



2

9

+

2

Z6

( 2a

( C)

= ( Z ) + 2( Z  ) ( ) + ( )



( 2a − b)

+

( 2a) 4a2

6Z 

+

2

− −

2

2

+ 9

2( 2a) ( b) + ( b) 12ab + 9b2

2

2

2

− (y) = ( 2a ) − 2( 2a ) ( (y) + ( (y) 4a6

(

−

2,ay

+ 2(y2

PRACTIHUEMOS

E0e%*/$ : 3(

 x = 12

J

5(

m > +2

J


@(

 x 4 = 2

J

(

4 x = y 2

J

(

2a > b2

J

(

a = +b2 J

6(

 x D > 62

4(

2a = 6b2 J

(

2a = 4b42

3F(  x 4 >  y 2

J

J

J

PRACTIHUEMOS EN CASA Res'l8e : 3(

 x = +b2

5(

D x = 112 @(

(

 y = 42

(

6 = a2

(

 x > 2

3F( m  42

6(

(

a2 = 1,b2 2a > b22

4(

m = n2

1,m > + y (2


II(

CUBO DE UN BINOMIO 3

( a + b ) = a3 + 3a2b + 3ab 2 + b 3 3

( a − b ) = a3 − 3a2b + 3ab 2 − b 3

Ee.5l's : ( y + )



3( ( m− 4)

5(

C

( y)

=

( m)

=

mC

( 4 x − 2y)



@(

= =

( x





2

( y) ( ) + ( y ) ( )  = y + 9y2 + 2Dy =

+

C

C( m)

−

2

−

16 x2 −

2

+

+

( 4) + C( m) ( 4)

12m2

−

( 4 x)



2

2

+

C( 4 x)

2

4+m

( )



2D

2

− −

( 4)

C

64

( 2y) + C( 4 x) ( 2y)

96x2y

+

4+ xy2

2

2

−

−

2

( 2y ) +y 

+ 2b) = ( x ) + ( x ) ( 2b) + ( x ) ( 2b) + ( 2b) = x6 +

(

6

6x b

+



2

12x b





+ +b

PRACTIHUEMOS

E0e%*/$ : 3(

4 x = ( J

5(

a =  J


@(

2m = n J

(

a2 > 2b J

(

2a = x  J

(

1 = x  J

6(

a2 = + J

4(

a2 = 2b J

ALGEBRA (6to grado

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