I.E.P. “DOSCIENTAS MILLAS PERUANAS”
ÁLGEBRA b IDENTIDADES DE LEGENDRE%
Ta*b+,% (a " b " ' ! a"b"' " (a " b(b " '(' " a
III. POLINOMIOS(continuación)
) ( – ( " 4 !
(a"b"' ! (a"b"' (a2"b2"'2 - 2(a"b"'" )ab'
PRODUCTOS NOTABLES PRINCIPALES 1.1.
3.
PRODUCTOS NOTABLES
8.
(I/$ (I/$+/a +/a/ / / / S$ S$+ +
Ca6'76a % a 2 " b2 %
("a("b("'! " (a"b"' "(ab"b'"'a " ab'
(a - &), / a, - &, - ,a&
PROBLEMAS RESUELTOS
Coolaio ! "Id#ntidad#$ d# L#%#nd#"
-
(a"b2 " (a-b2 ! 2(a2"b2 (a"b2 - (a-b2 ! &ab
2
(a"b"' ! a "b "' "2ab"2b'"2'a
2
2
(a"b ! a "a b"ab "b ! a "b " ab(a"b
(a-b ! a-a2b " ab2 - b ! a -b-ab(a-b
#) Suma Suma Di'##n Di'##ncia cia d# Cu&o Cu&o$$ 2
2
2
(a"b(a -ab"b ! a "b 2
+, - 10+ - ,
9 / (811), - ,(811)(8) - (),
Solución !
Tino inomi mio o al al Cu& Cu&o o
"
(a (a
-
(a "12 – (a -12 % .....................................
+, 5, /
4.
(9 " ) (9 – )
% ...... ......... ...... ...... ...... ...... ...... ...... ...... ...... .....
10.
(a " 1 (a – 1
% .......... ................ ........... ........... ............ ........
11.
(* – &(* " &
% .......... ................ ........... ........... ............ ........
12.
(2 " (2 –
% .......... ................ .......;;;. .;;;...... .....
1.
( – ( "
% .......... .............;;; ...;;;..... ....... ..
PROBLEMAS PROPUESTOS +, 64
&( (2 " (2 –
I.
R#$u#l:# lo$ ;oducto$ nota&l#$ !
( " 8 ("10 ! Solución ! +, - (7-10) + - (7) (10) / +, - 17+ - 70
( "
% .......................;;;..;.......
1.
% ..............................................
13.
( –
(9 – 2
% ..............................................
(* " 2
% ..............................................
$i%ui#nt#$
a BINOMIO AL CUADRADO%
Solución ! (,+), , / 6+, ,
1&.
1).
Solución !
(a"b"' ! a"b"'"a2(b"'"b2('"a"'2(a"b ")ab'
9 / 1,1 - ,, - ,
("3 (-3
(a-b(a "ab"b ! a -b
')
2
"
/ BINOMIO AL CUBO%
Solución !
+, ,(+) (2) - 2, / +, 3+ - 4
d) Bino Binomi mio o al Cu&o Cu&o
15 / a, - &,
2 ( – 2
2
(a
8 ! (-11 (-11 – 92
+, - ,( + )( ) - , /
c) Tinomi inomio o al Cuada Cuadado do 2
, 7 / a, - &,
Resuelve utilizando productos notables :
Solución !
(a"b(a-b ! a2-b2
2
, / a, - &, - ,(6)
1 ( " 2
&) Di'## Di'##nci ncia a d# Cuada Cuadado$ do$
-&2
(
' DI:ERENCIA DE CUADRADOS%
Solución !
2
N#$a% (a-b 2 ! (b-a 2
"
3 S+ a " b ! 5 ab ! & ("a("b ! 2"(a"b "(a"b"a "ab b
a) Bino Binomi mio o al Cua Cuad dad ado o (a b2 ! a2 2ab"b2
&2
"
2 % ......................................
+, - 3+ ,5
%) Pod Poduc ucto to d# Bino Binomi mio$ o$ con un Tmino Com*n
(
% .......................................
+,- (82-4) + - (82)(4)
(a"b"' ! a "b "' " (a"b"' (ab"b' " 'a - ab'
1.
).
Solución !
1.
(a " 32 % ............................................
2.
(* " 2 % ...........................................
.
(2"22
% ...........................................
&.
(* -42
% ...........................................
.
( - )2
% ...........................................
/ BINOMIOS CON UN T
( " )( )(" " % ...... ......... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ..... ..
14.
(-( (-("3 "3 % ...... ......... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ..... ..
20.
(* " (*-8 (*-8 % ...... ......... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... .....
2 1. 1.
( " 2 ( ( " % ; ;; ;; ;; ;; ;; ;; ;; ;; ;; ;; .
2 2. 2.
( a – 2 (( a - & % ; ;; ;;; ;; ;; ;; ;; ;; ;;; ;
I.E.P. “DOSCIENTAS MILLAS PERUANAS”
/ 0 10 (
–
(
"
&
1 (
3
-2
3
(
"2
!
S+*6++'a % 3
( 2
( 10
3 ( 10
3
(
!
2 (
a / –20
(3"2(3-2
!
2
2 ( )
2 ( 2
b 12 -12
&
(
1(
1 ( 2
' -13
1( 2
a )
1
8
(
&2
"
) (2
!
12
"
!
a –1 / –)
1(
1 2( 2
b 2 -3
1( 2
' -
-
2
E'$a %
a 8 / 20
! &
8 (&*
4 (
"
"
)2
(
"
!
3
b 10 2&
' 1)
' 2
!
2
' 2
(2-1(2"1(&2"1(1)&"1"1-2) 8 b
3
8 '
4 '
10a
11/
/ &
,.
a 1
1
4
b a
' a8
/ a&
-1
E'$a %
a –1& / –20 10
b –1) -22
DI
,.1.
DE>INICI=N !
EH 6a #a'+ 6a '7a6 /a/aH /#H H+#H a6Jba+'aH 66a*a/aH /++//# 9 /++H#K '#H+H$ #b$ #$aH /#H 66a*a/aH '#'+$ 9 H$#K '7*6+,/#H 6a H+J7+$ 6a'+.
' –18
R ! 0 D. Ea'$a D ! F/ " R R 0 D. Ia'$a D ! D++//# / ! D++H# F ! C#'+$ R ! RH+/7# # H$#
R/7' %
11
' 1
,.,.
E'$a %
( " &(& " – 20 ( " 1 2 a – b – ' 0 /
E'$a %
a
) a
( " 2 - ( " 2 2 " ( " &2 - ( " 2 a –& b – ' –2 / –1 0
E'$a % ( " 1 " ( – 1 – 2 a b ) / & N.A.
'
/
( " 1( " 2 - ( " 2 " ( - 2 - ( - &( -
($ " 2($2 - 2$ " & - ($ - 2 ($ 2 " 2 $ " & 3 (29
b 2
A ! (a-1(a"1(a2"1(a&"1"1
(9
2
a66a 6 F7+a6$ / %
9
!
2 a
& b
a66a 6 a6# / % P ! 2 " 92
Ca6'76a 6a /+'+a $ % (
1 '
Sab+/# F7 % " 9 ! 85 9 ! &
2 (
S+ % a " b ! ab ! 2 Ca6'76a % a2 " b2 a 21 b 13 ' 13 / 13 2
II. Subraya la alternativa correcta : 1
CLA
! )
II. Desarrolla los siguientes Productos Notables :
ÁLGEBRA
-1
PROPIEDADES DEL ?RADO !
1 E6 Ja/# /6 /++//# /b H *a9# # +J7a6 F7 6 Ja/# /6 /++H#. >? D @ >? / @
I.E.P. “DOSCIENTAS MILLAS PERUANAS” 2 E6 Ja/# /6 '#'+$ H +J7a6 a6 Ja/# /6 /++//# *#H 6 Ja/# /6 /++H#.
E6 Ja/# /6 H+/7# aa /H/ '# (0 aH$a 7 Ja/# *#H F7 6 Ja/# /6 /++H#. > ? R @ M+*# ! 0
a) MONOMIO ENTRE MONOMIO !
2
3 x y
4
! &&9
3
6
y
5
4 2
2 x &
8
x y
16
y
2 )
Pasos a seguir : 1 C#+'+$ /6 /++//# #/a/# /''+$*$ 7a a+ab6K '#*6$# # '#*6$a/#.
6 C#+'+$ /6 H+/7# F7 H #b$+ / H7*a 6aH '#67*aH +a6H 7a #b$+/#H $#/#H 6#H '#+'+$H /6 '#'+$.
&) POLINOMIO ENTRE MONOMIO ! 8 x
M,$#/# / R7++.
2 C#+'+$ /6 '#'+$ F7 H #b$+ / /++/+ 6a H7*a / 6#H 6*$#H / 'a/a '#67*a $ 6 +* '#+'+$ /6 /++H#. Ca/a '#+'+$ /6 '#'+$ H *76$+6+'a # 6#H /*H '#+'+$H /6 /++H# aa '#6#'a /+'#H H76$a/#H a a$+ / 6a H+J7+$ '#67*a #*a #+#$a6.
CASOS !
12 x y
, C#+'+$ /6 /++H# #/a/# /''+$*$ 7a a+ab6K '#*6$# # '#*6$a/#K '# H+J# '#$a+#K Ha6# 6 +*#.
> ? R @ M+*# ! >? / @ - 1
6
M,$#/# / #.
a) M@TODO DE ILLIAM ?. ORNER !
>? F @ ! >? D @ - >? / @
,.2.
3
M,$#/# N#*a6 ('6H+'#.
M,$#/# / C#+'+$H Haa/#H.
6 RH$# / 6a /++H+ F7 H #b$+ / H7*a 6a 6$+*a '#67*a. ESUEMA ?ENERAL !
PROBLEMAS RESUELTOS
1.- D++/% 2)941 1)94 Soluci)n: 2) 1) 94 -41
1
2
1)1
&
2.- E'$a %
.1. TEOREMA DEL RESTO! &
LQNEA DIISORA
M@TODOS !
2 L#H '#+'+$H /6 '#'+$ F7 H #b$+ / H7*a 'a/a '#67*aK 67J# F7 6 '#+'+$ a$+# H a *76$ +6+'a/# # 2 9 '#6#'a/# 6a H+J7+$ '#67*a.
&SER'#CI(N: S+ 6 '#+'+$ +'+a6 /6 /++H# H /+$ / 6a 7+/a/K 6 '#'+$ #b$+/# H /b /++/+ $ H$ a6#.
2
,.6.
, E6 a6# F7 H #b$+ aa 6a a+ab6 '7a/# 6 /++H# H +J7a6a a '#.
c) POLINOMIO ENTRE POLINOMIO ! *CRITERI !"ND#$ENT#%.L#H #6+#*+#H /++//# 9 /++H# /b H $a '#*6$#H 9 #/a/#H /H'/$*$ H'$# a 6a +*a a+ab6.
1 L#H '#+'+$H /6 /++//# /b H$a #/a/#H /''+$*$K '#*6$#H '# H'$# a 7a a+ab6.
1
2
! & 9 – 8 9 " )9
Pasos a seguir :
ESUEMA ?ENERAL !
5
x y
12
&) M@TODO DE PAOLO RU>>INI! S 7$+6+a '7a/# 6 /++H# H / +* Ja/#.
ÁLGEBRA #b$+/# Ha *# F7 6 Ja/# /6 /++H#.
&SER'#CI(N : La 6a /++H#a H '#6#'a Haa/# $a$#H $,*+#H / 6a a$ +a6 /6 /++//# '#*# Ja/# /6 /++H#.
S 7$+6+a aa #b$ 6 H$# / 7a /++H+. C#H+H$ +J7a6a a '# a6 /++H# 9 /Ha 6a *a9# #$'+a / 6a a+ab6K aa F7 Ha *6aa/a 6 /++//#. &SER'#CI(N: DH7,H / a6+a 6 *6a#K /b '#*#baH F7 6 Ja/# /6 #6+#*+#
18
5
2 x
10
2 x
Soluci)n : 2 .18
2 .10
.- ( 5
9
27 x y
8
. 8
15
3 x
4
y
7
Soluci)n: 5
9
8
4
7
27 x y
15
3 x y
15
3
15
3
&.- D++/%
5 y
x
5. 3
5
15
x y
3 3
5
x y
I.E.P. “DOSCIENTAS MILLAS PERUANAS” (3 – 10 " 12 (-2
( ! &2 " ) " & R( ! 10 "
6 x
Soluci)n: 7
5 x
2
3
x
10
5 x
2
2
5 x
15 x
5 x
- " 2 – .- D++/% 27a
7
b10
a 28b15 63a10 b11
36
9a
3
b4
Soluci)n: 27a 9a
7
3
b10
b4
36a
28
9a
3
b15
b4
10
63a
9a
3
b11
b4
a&b) " &a2b11 – 3a3b3 ).- D++/% 8. &
). 2
0. 2.
2
E6 '#+'+$ /6 $,*+# 6+a6 / R( H% 10 La H7*a / 6#H '#+'+$H
). 2
. 1
(
2
-12 " 18 - ) 8
2
2 1
- 2 " 3 10 "
0
-)
&
3 2
-8 12 -&
&
-
R(
4
'#'+$ H% , - 1 - 81 x
3 /
1
D #*a H+*+6a H #b$+ aa 6 H+/7#.
4
8.- E'$a % (8& - )2 " & " 3 (- " 1 " 2 2
&
Soluci)n : (8& – 0 – )2 " & " 3 (22 – " 1
-1
1 3
2
1
3
1&
3
2
1&
(
3
-1
1
3
R( !
5 3
x
2
14
3
19 x
3
-
-8 12 -&
(9 ! &2 " ) " & R( ! 10 "
O&$#:acion#$ ! 3.- E'$a 6a H+J7+$ /++H+ %
OBSER
H % ,+2
E6 $,*+# 'b+'# /6 '#'+$
E6 $,*+# '7a/$+'# /6 '#'+$ H % +, E6 $,*+# 6+a6 /6 '#'+$ H % 8+ E6 $,*+# +//+$ /6 '#'+$ H%852 -
0 12
)
-) -& 18 &
&
3
-) 12 -& 10
( ! &2 " ) " & R( ! 10 "
-
) &
8
&
-
-12 18 -)
10
2 -1
C#*6$a % ( ! 2 " 2 – -
12 -10 & 8 -2 3
-82 " 12 - &
x
2 x 1
14
8
12 - 102 " &
2
&. 3
&2 " ) " &
3 x
10 - 2 / 12
8& " 0 - )2 " & " 3 2 2 – " 1
-2
-1
$todo Coe,icientes Separados:
-8& " 12 - &2
1
-2
/ R( H%
$todo Nor+al :
x
2
La H7*a / 6#H '#+'+$H
3
2 x
6+a6 / S(. H % 3
6 - 3 - 6 / 16
. 1 2 . 2
3
E6 $,*+# '7a/$+'# /6 '#'+$ H % 6+, E6 $,*+# 6+a6 /6 Soluci)n ! '#'+$H H % 3+ E6 $,*+# +//+$ /6 O/a*#H 9 '#*6$a*#H % '#'+$ H% 6 E6 $,*+# 6+a6 / R( 6 5 4 3 2 6 x x 0 x 2 x 3 x H % 10+ 3 2 E6 $,*+# +//+$ 3 x x 2 x 1 / R( H % 2 E6 '#+'+$ /6 $,*+# ) 1 0 -2 -1 '7a/$+'# / ( H % 6 2 E6 '#B+'+$ /6 $,*+# 1 -& -2
&. 3
Soluci)n : O/a*#H 9 '#*6$a*#H % 8. &
5 x
3 x
/ ( H %
6
2
ÁLGEBRA E6 '#+'+$ /6 $,*+# 'b+'# /6 '#'+$ H % , E6 '#+'+$ /6 $,*+# '7a/$+'# /6 '#'+$ H % 1 E6 '#+'+$ /6 $,*+# 6+a6 /6 '#'+$ H % 81 La H7*a / 6#H '#+'+$H /6 -
-
E6 $,*+# '7a/$+'# /6 '#'+$ H % 6+, E6 $,*+# 6+a6 /6 '#'+$ H % 3+ E6 $,*+# +//+$ /6 '#'+$ H% 6 E6 $*+# 6+a6 /6 H+/7# H % 10+ E6 $*+# +//+$ /6 H+/7# H % 2 C#+'+$ /6 $,*+# '7a/$+'# /6 '#'+$ H ! 6 C#+'+$ /6 $,*+# 6+a6 /6 '#'+$ H % 3 C#B+'+$ /6 $,*+# 6+a6 /6 H+/7# H%10
La H7*a / 6#H '#+'+$H /6 '#'+$ H% 6 - 3 - 6 / 16
I.E.P. “DOSCIENTAS MILLAS PERUANAS” Soluci)n : La H7*a / 6#H '#+'+$H /6 H+/7# H% 10 - 2 / 12 P# #% & & 1 - – -) 4.- D++/% 2 x
4
5 x
3
2 x
2 x
2
2
28 –) –10
4 x 8
2
12
–12 -1
–18
1
Soluci)n :
2
-2
-)
2
-1 2
-2
-1
2 -2
2
-1
1
&
8
2
-2 )
........................................................................ ........................................................................
II. Resuelve :
1 (-1)& (2 !........................................
1.- a66a 6 H+/7# %
2 (-89 (-&9 !........................................ (&010 (3 !.......................................
(
x
2
.
&
&.
7
8
x
2
2
.
& (1 9 (-9 !.................................. x
2
).
2
b &-1
) (-&2a b ' (-3ab' !...........................
/ 10-2
1-11
........................................................................ 11 (-100a10 " 80a3 (a ........................................................................ 12 (20.&0980O100-)0.3090"1).1918T(-&.10918
R$a % 2
........................................................................
.
........................................................................
&.
2
28 .
2
.)
2 . 12
1& (1&&9&& - ))9)) - 388988(-22922 ........................................................................
' 10
12
x
8
x
' 1-10
a
b -1
' "2
7
3 x
5
x 1
1
/ "1 –
3.- Ca6'76a 6 H$# %
.
&
.
10 .
.
2
2
10 .
. 1
. 1
4 x
15
x
12
x
10
I/+'a 6 '#'+$% x
a 2--1
b 2"2"1
' 2"1
/ 2-2-1
2"2-1
.- D++/ %
1 (1)a3b '8 " 18a )b 10'4 - 1&a4b12 T (-2a &b
11.- A6 '$7a %
b –1 &
2 x
2.- D++/ %
10 (-188 " 213 - 4 (-2
( " 3 " (2 – - 38 – - 2 (-2"3 " ?(-22 – (-2 – 3@8 – (-2-2 13 " (-18 " 2 – 2
I/+'a H7 H+/7#%
. & 1) . 8 . 2 .
x
4 (&8*493 (1)*39& !.............................
R*6aa*#H %
' -8
.- I/+'a 6 H$# %
3. 2
a 1"11
)
8 (138988 (12 29& !........................
"2!0 ! -2
1 .
b –1) &
2 . 1
3 (-1&&292 ()1912 !.......................
Soluci)n :
a 1& / )
).- Ca6'76a 6 H$# %
(2&*1020 (-8*411 !........................... 8 3
2 . 12 .2
a –2 / 1
I. E,ect-a : / Pt. c0u1
3 8
13 ( " 1 " - ( " 2
10
x
&
&.
2
PROBLEMAS PROPUESTOS
& 1 4
10.- Ca6'76a 6 H$# %
&.
8
........................................................................
( ! 2 " 2 – 1 R( ! 4 " )
( x 3 7
)
14 (22"3-12-20&")-1 (1"2-
RH+/7# ! -2 - )
2
........................................................................
18 (). - &. " ) - . ) - 2. - 3. 2 T (2 " . & - . 2
2
x
ÁLGEBRA &.- I/+'a 6 H+/7#%
1) ( " 2& " ) - ( - 2
-
2.
.
&
. I/+'a 6 '#'+$%
a --1
.
)
2.
1 .
b 2"&-1
1 ( " ) - " ) ( " 1
' "22"&-1/ 2"2-1
........................................................................
"2"1
a 2
b
' 12
2 x
7
3
x x 1
1
/ )
2
I.E.P. “DOSCIENTAS MILLAS PERUANAS”
PROBLEMAS RESUELTOS CLA
8 *4
) )a3b&'
3 -2&12920
8 23)9&
4 * 9
10 ) -3 "
2
)
11 -20a"1)a2 1 -8a'8-4a2b'4"3ab3
1. >ACTOR PRIMO ! EH aF76 a'$# # '#H$a$ F7 $+ '#*# +'# /++H# a #$a H+ +/,$+'a a 66a. E*6#H %
,. M@TODOS !
1& -32922"28&9&&"4)9)) 1 ( ! )&-)"2-") 1) ( ! ")&"1&"22"0"100 R(! 20
,.1 >acto com*n o a%u;ación d# tmino$. ,., >actoiGación ;o id#ntidad#$.
13 ( ! )-&"102-2-2 R( ! &"0 18 ( ! -2-2-
I<. >ACTORIFACI=N.
" 1 a'$# 6+a6 2 " 1 a'$# '7a/$+'#
12 -09)2100"1)0912-&
a) Tinomio cuadado ;#'#cto ! R( ! -1&2"12
A2 2AB " B 2 ! (A B2
14 ( ! 2-)2-3"8 R( ! -1 II. 1 a & b 3 b
2 b /
' ) b
ÁLGEBRA
A& " B " C2 " D " E
1.- :a'$#+a%
A B!(A B(A2 " AB " B2 ,.2 Mtodo d#l a$;a $im;l# ! A2* " B*9 " C92 a) A$;a do&l# ! A2"B9"C92 " D " E9 " : &) A$;a do&l# #$;#cial !
Soluci)n: AHa H+*6%
Soluci)n: 2( " 2 " " 1 ( " 1 2
$2 " 8$ " 1 $ $
($" ($" 2.- :a'$#+a% a2b - b Soluci)n: b(a2 - 1 b(a " 1(a - 1
3.- :a'$#+a % &9 " 10&9 – 9 " 92 " )9 Soluci)n : :a'$# '#* % 9(&2 " 10 – 9 2 " 9 " )
.- :a'$#+a% 3a92-b"3b92-a Soluci)n: 3a92 " 3b92 - b - a
aHa /#b6 9(2 " 9 " 2 (2 – 9 "
392(a " b - (b " a
&) Di'##ncia d# cuadado$ !
c) Suma o di'##ncia d# cu&o$ !
).- :a'$#+a% 8$ " $2 " 1
" & " " 2
(a " b(392 - A2 – B2!(A " B (A - B
($ - '2
&.- :a'$#+a% 4 - &a2b&') Soluci)n: D+'+a / '7a/a/#H% ( " 2ab2'( - 2ab2' .- :a'$#+a% 4$2 " '2 - )$' Soluci)n: 4$2 - )$' " '2 $ –' $ –' L7J#% ($ - ' ($ - '
8.- :a'$#+a % ) – 9) Soluci)n : D+'+a / '7a/a/#H %
( " 9 (- 9
P# +/$+/a/H % ("9 (2 – 9 " 9 2 (-9 (2 " 9 " 9 2 ("9 (-9 (2 – 9 " 9 2 (2 " 9 " 9 2
4.- RH76% 2 "12 ") ! ( " ) ( " ) ! ( " ) 2
I.E.P. “DOSCIENTAS MILLAS PERUANAS” 10.- :a'$#+a 3
a&b& " 2a8b
- 0a4b&
1)
! .......................
22 – V – – V 2 "3V – 2 Soluci)n:
8
O/a*#H%
( – 9 a " (-9 b
13
1)
2)
! ................................................
1
V
-2
'2
23
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I.E.P. “DOSCIENTAS MILLAS PERUANAS” &) " 49& – 1 292
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