The document is a collection of algebra exercises involving notable products and quotients solved by engineer Widmar Aguilar. It contains 25 problems involving steps to solve expressions and equations related to factoring, multiplying, dividing, and simplifying algebraic terms. The problems progress from simpler expressions to more complex equations involving radicals, fractions, and multiple steps. The document provides the work and solutions to each problem.

1. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
EJERCICIOS RESUELTOS DE ALGEBRA
PREUNIVERSITARIA
PRODUCTOS Y COCIENTES NOTABLES
Ing. WIDMAR AGUILAR, Msc
Febrero 2021

2. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
Se puede utilizar la siguiente información:

3. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.

4. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.

5. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.

6. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
Además:

7. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.

8. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
1)
De;
= → ( + ) = 4
+ 2 + = 4 → − 2 + = 0
( − ) = 0 → =
=
√
+
=
√
+ = + 1 = 1 + 1
= 2 → ( )
2)
De:
+ = 1 ; ( + ) = 1
+ 2 + = 1 → + = 1 − 2 … … . (1)
( + ) = 1 ; + 3 + 3 + − 1
+ = 1 − 3 ( + )
+ = 1 − 3 (1)
+ = 1 − 3 … … … … . . (2)
Reemplazando (1) y (2) en la expresión a calcular:
= 6( + ) − 4( + )
= 6(1 − 2 ) − 4(1 − 3 )
= 6 − 12 − 4 + 12
= 2 → ( )

9. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
3)
Multiplicando entre sí las expresiones:
Eb = √ + + √ − ! √ + − √ − !
= "( + ) − ( − )#
= + − + = 2
= 2 → ( )
4)
Sumando 2 a cada miembro de:
$ + $ = 7 → $ + 2$ = 5
($
'
+ $
'
) = 9 → $
'
+ $
'
= 3 ----(1)
Sumando a la expresión (1), -2 a cada lado:
$
'
− 2 + $
'
= 3 − 2
$
'
− 2 + $
'
= 1
()$
'
- )$
'
) = 1
*$
+
− $
+
, = 1 → -)

10. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
5)
.
+
.
= → =
( + ) = 4 → + 2 + = 4
− 2 + = 0 → ( − ) − 0, 01234:
= --------(1)
Reemplazando (1) en E;
= $
( )678
678 678
6
= $
( )678
678
6
= $
. 6. 678
678
6
= √29
6
= 2 → ( )
6)
( + ) = 1 → ( + ) = 1
( + + 3 + 3 ) = 1
( + ) + 3 + 3 = 1
Como E = ( + )
+ 3 ( + ) = 1 → + 3 :
.
; = 1
= 1 − 3 ……………………(a)
De :
= ( + ) = ( + )( − + )
= (
.
)( ( − + )
= ( − + ) = ( + ) −
= ( + ) −
= 2 − --------(b)

11. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
Igualando las ecuaciones (a) y (b):
1 − 3 = 2 −
= −0,5
Reemplazando en (b),
= 2 − (−0,5)
= 2,5 → ( )
7)
<
− = 6(2 − )
< = =
= 6(2 − ) → 8 − = 6 92 − ), factorizando:
(2 − ) (4 + 2 + ) = 6 (2 − )
4 + 2 + = 6
4 − 4 + = 0 → (2 − ) = 0
= 2 , luego:
E = 4( + )
= 4 : + ; = 4 :
.
+ 2;
= 4 : ; = 10 → ( )
8)
Desarrollando la expresión E;
= + ? + 3 ?( + ?) − 18( + ?) + 20
= + ? + ( + ?)(3 ? − 18) + 20
@A B-4 0 A ℎDEóG2ADA, A2 GD2B2:
= (10) + ( + ?)(3(6) − 18) + 20

12. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
= 10 + 20
= 30 → ( )
9)
+ = −
.
→ = −
.
4 + 4 = −
+ 4 + 4 = 0 → ( + 2 ) = 0
= −2
Utilizando E:
= +
<
= 1 + 2
= 3 → ( )
10)
$
=
+ $
=
= 3 ; 2; 202H B-4 0 1 4,
( $
=
+ $
=
) = 27
+ + 3$ .
=
($
=
+ $
=
) =27
+ = 27 − 3(3)
+ = 18
− 2 + = 18 − 2 → ($ − $ ) = 16
Sacando la raíz cuadrada de cada lado,

13. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
$ − $ = 4
= $ − $ = 4 → (2)
11)
( − ) = 3 ; ( + ) = 7
Elevando al cuadrado a ( − ) = 3 .
( − ) = 9
( − 2 + ) = 9
( + ) − 2 = 9, como ( + ) = 7:
7 - 2 = 9 → = −1 → = −1
= 1 ………………..(a)
Como E = ( + + ) , sumando y estando ,
= ( + + − + )
= "( + 2 + ) − }
= "( + ) − # ; J G4KDL B-4:
= ( + − )( + + )
Usando: → = −1 ? = 1 , se tiene:
= (1)(
M
+ 1)(
M
− 1)
= :
M
.
+ 1; :
M
.
− 1; = (8)(6)
= 48
12)

14. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
Reduciendo el numerador y denominador:
( + ) + + = + 4 + 6 + 4 + + +
= 2 + 2 + 4 + 6 + 4
= 2( + + 3 + 2 + 2 )
{( − ) + ( + ) + 2 } =
= ( − 2 + + + 2 + + 2 )
= (2 + 2 + 2 )
= 4( + + )
= 4" + + + 2( + + )#
= 4( + + 3 +2 + 2 )
=
( ' ' = =)
( ' ' = =)
=
.
→ ( )
13)
+ + = 3 ; -2A 4OE4BD2B-4 20 3 2B 1 ? 2;
+ + = 1 + 2
+ ( − 1) + ( -2) =0 ; elevando todo al cubo,
" + ( − 1) + ( − 2)# = 0
+ ( − 1) + ( − 2) = 3( )( − 1)( − 2) ….propiedad
=
= ( .)= (P )=
( .)(P )
=
( )( .)(P )
( .)(P
= 3
= 3 → ( )

15. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
14)
=
QR QS Q=
QS
=
Q=
QS
( − + 1) --------(1)
De:
− 3 + 1 = 0 → + 1 = 3 ; 202H B-4 0 1 -K -4:
( + 1) = 9 → + 1 + 2 = 9
+ 1 = 7 − − − − − −( )
( ) 2B (1):
=
Q=
QS
"( + 1) − )
=
.
Q
( 7 − ) =
.
Q
(6 )
= 6 → ( )
15)
= $
6 6
√ 6 6
=
= $
6 6
6 6
=
= $
6
6 6 +
6
6 6
=
= $( )
6
+ 2( )
6
=
--------(a)
De:
( )9
+ 4 ( )9
= 725 → ( )9
+ 4 + 4 ( )9
= 725 + 4
( )9
+ 4 + 4 ( )9
= 729

16. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
( )
6
+ 2( )
6
) = 27
( )
B
2 + 2( )
B
2 = 27 …………………( )
( ) 2B ( );
E = $( )
6
+ 2( )
6
=
= √27
=
= √3
=
= 3 → ( )
16)
Sumando 2 para obtener un trinomio cuadrado perfecto,
+ 2 + = 34 + 2
( + ) = 36
+ = 6 ; A1O B-4 − 2 G4- 0 2 EK2ADóB
− 2 + = 6 − 2 → − 2 + = 4
( − .
) = 4 → − .
= 2
− .
= 2 → (-)
17)
( + ) = 3 , 202H B-4 0 T1DBG ,
( + ) = 3
+ 5 + 10 + 10 + 5 + = 3
+ + 10 ( + ) + 5 ( + ) = 3 -------(a)
De: ( + ) = 3
= + + 3 ( + ) ------(1)

17. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
(1) en (a) :
+ + 10 (3) + 5 (27 − 3 (3)) = 3
ab = 3-2 =1
+ + 10 (1)(3) + 5(1)(27 − 3(1)(3)) = 3
+ = 243 − 30 − 90
+ = 123 → ( )
18)
De: a+b = 5 → ( + ) = 25
+ 2 + = 25 ; + 2(5) + = 25
+ = 15 − − − − − −( )
( + ) = 125 → + + 3 ( + ) = 125
+ = 125 − 3(5)(5)
+ = 50 − − − − − ( )
(a) y (b) en E:
= = = .U
=
.
U .U
=
.
→ ( )
19)
.
+
.
=
.
→ =
.
( + ) = --------(a)
+ + 2 = → + = −
Elavando (a) al cubo:

18. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
( + )V
= -------(b)
De: + = − b
( + ) = (− )
V
+ V
+ 3 ( + ) = −
V
+ V
+ 3 (− ) = −
V
+ V
= 2 -----(c )
(a) , (b), (c) en E:
=
( )W V( W W)
( )=
=
3 3
V(2 3 3
)
( )=
=
.. 3 3
3 3
= −11 → (2)
20)
+ + = 0 → X
+ = −
+ = −
+ = −
De:
( + + ) = 0 → + + = −3( + )( + )( + )
+ + = −3(− )(− )(− )
+ + = 3
De:
=
P
+
P
+
P
=
= = P=
P
=
P
P
= 3
= 3 → ( )

19. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
21)
− 1 = √2 → ( − 1) = 2
− 2 + 1 = 2
+ 1 = 2(1 + ) … … … … … . . ( )
+ 1 = 2 + 2
− 1 = 2 … … … … … … … ( )
Como = 1 + 2 → = (1 + 2 ) − − − − − ( )
=
YQ .ZYQW .Z
Q'
=
((Q .)((Q )= .)
Q'
=
YQ .ZYQ .Z(Q' Q .)
Q'
=
(Q .) "Q' YQ .Z#
Q'
; reemplazando (a), (b) y (c)
=
(Q .) "(. Q) Q#
(. Q)
=
(Q .) (. Q Q Q)
(. Q)
=
(Q .) (. Q Q )
(. Q)
=
(Q .) (. Q Q )
(. Q)
=
(Q .) (Q Q )
(. Q)
; = 1 + 2
=
(. Q Q )( Q )
(. Q)
=
(Q Q )( Q )
(. Q)
=
U(Q )
(. Q)
= 40.
(. Q)
(. Q)
= 40 → (-)
22)
+ = 62 − − − A1O B-4 2 - 0 -4,
+ 2 + = 64
( + ) = 64 → + = 8 √

20. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
= :
√
;
8
=
= :
<√
√
;
8
=
= (8)
8
= = √2
=
= 2
= 2 → ( )
23)
( + 2 + 2 + 1)( − 2 + 2 − 1) =
= V
− 2 + 2 − + 2 − 4 + 4 − 2 + 2 − 4 +
4 − 2 + − 2 + 2 − 1
= V
− 1 + (2 − 2) + (4 − 4) + (5 − 5) + (2 − 2) +
(2 − 2)
= V
− 1
= $
(Q= Q Q .)(Q= Q Q .)
Q .
+
= $
QW .
Q .
+ = $
QW Q' Q .
Q .
= $
(QW Q ) (Q' .)
Q .
= $
Q (Q' .) (Q' .)
Q .
= $
(Q' .)(Q .)
Q .
= $
(Q .)(Q .)(Q .)
Q .
= [( + 1) = + 1
= + 1 → ( )
24)

21. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
= ( V
+ 2 ? + V
? + ? − 2 ? + V V
).
.
' W
+
2 ?
= ( V
+ V
? + ? + V V).
.
' W
+ 2 ?
=
Y 'QW ''Z Y W' WQWZ
' W
+ 2 ?
=
'YQW 'Z W(QW ')
' W
+ 2 ? =
YQW 'Z( ' W)
( ' W)
+ 2 ?
= ( V
+ ? ) + 2 ? = V
+ 2 ? + ? , , , , , , , ]^_
= ( + ? ) → ( )
25)
= : + + + − − ( + + − + );. ( +
+ + − + + + − + )
= ( + + + − − − − + − )(2 +
2 + 2 )
= (2 − 2 )( 2 + 2 + 2 )
= 4( − )( + + )
= 4 ( + + − − − )
= 4 ( − )
= 4 − 4 → ( )
26)
+ ? = → ( + ?) =
+ ? + 2 ? = ; ? = --------(c)
+ ? = − 2 − − − − − ( )

22. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
De:
Q= =
Q(Q )
=
.
, A2 GD2B2:
(Q )YQ Q Z
5 ?( +?)
=
.
→
Q Q
Q
=
.
--------(b)
(a ) y (c ) en ( b ):
=
.
→ − 2 − =
= 4
√ = √4 → = 2 → ( )
27)
= ( − + ) − ( − − )
= − + + 3( − )( + )( − ) − " − − + 3( −
)( − )(− − )
= − + − + + + 3( − )( + )( − ) − 3( − )( −
)( + )
= 2 + 3( − )( + )( − ) + 3( − )( − )( + )
= 2 + 3( − )"( + )( − ) + ( − )( + )#
E = 2 + 3( − ) ( − − + ) + ( + − − )
= 2 + 3(a-b) ( − − + + + − − ))
= 2 + 3(a-b) ( 2 − 2 )
= 2 + 6( − )( − ) = 2 ( − − + )
= 2 + 6( − 2 + )
= 2 + 6 ( −2ab + ) -------------(a)
De: : − 3; = : − 3;
: ; = : ;
( − 3 ) = ( − 3 )

23. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
− 3 = − 3
− = 3 ( − )
(a-b)( + + ) = 3 ( − )
+ + = 3
+ = 2 − − − − − ( )
(b) en (a), se tiene:
= 2 + 6 ( + − 2 b )
= 2 + 6 (2 − 2 )
= 2 → ( )
28)
= ? .
+ ? .
=
Q
+
Q
=
Q= =
Q
=
(Q )YQ Q Z
Q
− − − ( )
`
+ ? = 30
+ ? = 6
---------(1)
+ ? = 6 → ( + ?) = 36
+ ? + 2 ? = 36
30+2xy =36
xy = 3 -------(b)
( )? (1)2B ( );
=
V( U )
=
M∗V
= 54
= 54 → (-)

24. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
29)
+
.
Q
= 3 →
Q .
Q
= 3 → + 1 = 3 -------(1)
=
QW .
Q(Q' .)
=
(Q )= .
Q(Q' .)
=
(Q .)(Q' Q .)
Q(Q' .)
-------- (a)
+ 1 = 3 → ( + 1) = 9
+ 2 + 1 = 9
+ 1 = 7 ----------- (b)
( ) 2B ( ):
=
(Q .)((Q' .) Q )
Q(Q' .)
=
(Q .)(7 2
− 2
)
Q(7 2)
( 1) en E:
=
Q(VQ )
Q(7 2
)
=
.<
M
→ ( )
30)
b
( + ) = 1
( + ) =
Dividiendo entre si las dos ecuaciones anteriores,
= =Y = =Z
( )
= → ( + ) = ( + )
( + )( − + ) = (a+b)
( − + ) =

25. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
( + ) − =
( + ) = +
Como : = ( + )
= + − − − − − − − − − − − ( )
De:
( + ) = 1 → ( + ) = 1
( + + 3 ( + )# = 1
( + + 3) = 1 → + =
.
= = − 3 ---(b)
De: ( + ) = → + = = =
− − − −( )
( ) = ( ) →
.
= =
− 3 = = =
= = = −3 → = −
.
− − − −(-)
(-) 2B ( ):
= + = −
.
+
= 2 → ( )
31)
+ √ = + √
− = √ -√
Y√ − √ Z(√ + [ ) = √ -√ − − − −-DJ2K2B D -2 1 -K -4A
Y√ − √ ZY√ + √ Z = √ √ − √ √
Y√ − √ ZY√ + √ Z = √ Y√ − √ Z
Y√ − √ ZY√ + √ Z = −√ Y√ − √ Z
√ + √ = −√

26. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
√ + √ + √ = 0 − − − − − ( )
( ) − − − − − 202H B-4 0 1 4:
Y√ + √ + √ Z = 0
(√ ) + (√ ) + (√ ) = 3√ √ √
(√ ) + (√ ) + (√ ) = 3√ − − − − − −( )
De E :
=
√ P
+
√ P
+
P
√
=
√
√ P
+
√
√ P
+
P√P
√ P
=
√ =
√ P
+
√ =
√ P
+
√P=
√ P
=
√ 3+[ 3
+√ 3
√ P
( )2B ∶
=
√ P
√ P
= 3 → ( )
32)
( + ) = √3
=
→ ( + ) = 3
+ + 3 ( + ) = 3 − − − −( )
( − ) = √2
=
→ ( − ) = 2
− − 3 ( − ) = 2 − − − −( )
`
+ + 3 ( + ) = 3
− − 3 ( − ) = 2
( ) + ( ):
2 + 3 ( + − + )=5
2 + 6 = 5
2 ( + 3 ) = 5 → + 3 = − − − ( )

27. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
( ) − ( ) :
`
+ + 3 ( + ) = 3
− + + 3 ( − ) = −2
2 + 3 ( + + − )= 1
2 + 3 (2 ) = 1 → 2 + 6 = 1
2 ( + 3 ) = 1 → + 3 =
.
− − − −(-)
( ) ? (-) 2B :
= 4 ( + 3 )( + 3 )
= 4 : ; :
.
;
= 5 → ( )
33)
d
− = √7
R
− = √7
R
− = 2√7
R
-------------(a)
( ) 2B :
=
( P)R ( P)R ( )R
MU
=
Y √M
R
Z
R
Y √M
R
Z
R
Y √M
R
Z
R
MU
=
R.M M M
MU
=
e.U
MU
= 13
= 13 → ( )
34)
+ = + → + = ( + ) … … . ( )

28. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
=
( = =) ( = = )
( )
=
= = = = ( )
( )
( ) 2B :
=
= = = = = =
( )
=
( = =)
( )
− − − −( )
De: + = ( + )
(a+b) ( − + ) = ( + )
− + =
− 2 + = 0 → ( − ) = 0
= … … … … … . ( )
( ) 2B ( ):
=
( = =)
( )
=
( = =)
( )
=
V =
=
= 6
= 6 → ( )
35)
+ + = 2( + )( + )( + ) … … . A1O B-4 G4-4:
= = P=
+ = ( + )( + )( + )+abc
+ + + 2 = 2"( + )( + )( + ) + #
Como :
( + )( + )( + ) + = ( + + )( + + ) − − − EK4ED2- -
+ + + 2 = 2 ( + + )( + + )
( + + ) − 3( + + )( + + ) + 3 + 2 = 2 ( + + )( +
+ )
1 − 3 (1)( + + ) + 5 = 2(1)( + + )
1 − 3( + + ) + 5 = 2( + + )

29. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
+ 5 = 5 ( + + ) -------(a)
De:
( ) 2B :
=
. P
P P
=
( P P)
( P P)
= 5 → ( )
36)
Si: + + = 0
( + + ) = 0 → + + = 3 − − − ( )
( ) 2B :
=
f f Pf ( = =)( = P=)( = P=)
V( = = P=) . P
Además:
( ) + ( ) + ( ) + 3( + )( + )( + ) =
= ( 3 +
3
+ 3)
=
( 3+
3
+ 3)
=
V( P) . P
=
( P)=
P
= (3 )
= 9 → ( )
37)
( + 1)( − 1) = − 1
=
( . )( ' .)( W = .)( W = .)
f .

30. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
=
( W .)( W = .)( W = .)
( f .)
De; ( − 1) (( ) − + 1) = ( e
− 1)
( + 1) (( ) + + 1) = ( e
+ 1)
=
( W .)( = .)( W = .)( = .)( W = .)
( f .)( = .)( = .)
=
( W .)( f .)( f .)
( f .)( = .)( = .)
=
( W .)( f .)
( = .)( = .)
=
( = .)( = .)( f .)
( = .)( = .)
= e
− 1
= e
− 1 → ( )
38)
+ = 2 → = 2
− 2 + = 0 → ( − ) = 0
= − − − − − ( )
1A B-4 ( ):
E = ( )g
+ ( )g
= ( )g
+ ( )g
→ = (1)g
+ (1)g
Como h ∈ j → = 1 + 1
= 2 → ( )
39)

31. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
=
(
8
k
8
l
) (
8
k
8
l
)
−
=
8
k kl
8
l
:
8
k kl
8
l
;
−
=
8
k kl
8
l
8
k kl
8
l
−
=
'
kl
− =
'mk l
kl
−
=
( )
− =
( )( )
( )
−
= − = = = 1
= 1 → ( )
40)
Q
Q
= 1 −
Q
-------------(a)
Q
Q
= 1 −
Q
---------------(b)
(a) por (b):
Q
Q
.
Q
Q
= :1 −
Q
; : 1 −
Q
;
1 = (
Q
)(
Q
Q
)
? = −(? − )(? − )
-xy = (? − )
− ? = + ? − 2 ?
+ ? = ? − − − − − ( )
( + ? ) = ? → + 2 ? + ? = ?
+ ? = − ? − − − − − − − (-)
( )? (-) 2B :

32. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
=
(Q )(Q' ')
QW W
=
(Q )(Q' ')
"(Q )= ( )=#
=
(Q )(Q' ')
(Q )(Q' Q ')
=
Q' '
Q' Q '
=
Q
Q Q
=
Q
Q
=
.
→ ( )
41)
( + + + -) = 4 ( + )( + -)
Agrupando el factor cuadrático:
"( + ) + ( + -)# = 4 ( + )( + -)
( + ) + ( + -) + 2( + )( + -) = 4 ( + )( + -)
( + ) − 2( + )( + -) + ( + -) = 0
"( + ) − ( + -)# = 0
( + ) = ( + -) − − − − − −( )
( ) 2B :
= 4
n7o
(k7l) = 4
k7l
(k7l) = 4
8
= √4
= 2 → (-)
42)
+ + + - = 0 ; + - = 1
a+b +1 =0 → + = −1
( + + 1) = 0 → + + 1 = −3
+ = −1 − 3 − − − − − −( )
( + - − 1) = 0 → + - − 1 = −3 -

33. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
+ - = 1 − 3 - − − − − − − − ( )
De: + = −1 → ( + ) = 1 → + 2 + = 1
+ = 1 − 2 − − − − − ( )
+ - = 1 → ( + -) = 1 → + 2 - + - = 1
+ - = 1 − 2 - − − − − − (-)
( ) , ( ) , ( )? (-) 2B :
=
3
+ 3
+ 3
+ -3
( + ) − − -
=
. . Pp
( ) P p
=
( Pp)
. (. Pp)
=
( Pp)
. . Pp
( Pp)
( Pp)
= → ( )
43)
+ ? + L = 4
( + ? + L) = 64
+ ? + L + 3( + ?)(? + L)( + L) = 64
+ ? + L + 3( ? + L + ? + ?L)( + L) = 64)
+ ? + L + 3 (16 + ? + L + L + ? + ? L + ?L + ? ) = 64
+ ? + L + 3" ?(4 − L) + ?(4 − ?) + ?L(4 − )# = 64 − 48
+ ? + L + 3 "4 ? − ?L + 4 L − ?L + 4?L − ?L ) = 16
+ ? + L + 3 "4( ? + L + ?L) − 3 ?L# = 16
Como:
? + L + ?L = 6 y xyz =8
+ ? + L + 3 "4(6) − 3(8)# = 16

34. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
+ ? + L = 16
E = + ? + L
= 16 → ( )
44)
+ ? = √5
'
+ 2? → − ? = √5
'
(q − r) =(√5
'
) → − 2 ? + ? = √5
? = √5 → + ? = √5 + 2√5
+ ? = 3√5
+ ? = 3√5 → + ? + 2 ? = 2 ? + 3√5
( + ?) = 5√5 → + ? = (5√5)./
+ ? = √5√5
'
+ ? = 3√5 → + ? − 2 ? = −2 ? + 3√5
( − ?) = √5 → − ? = (√5)./
+ ? = √5
'
De:
( + ?)( − ?) = √5√5
'
. √5
'
− ? = Y√5ZY √5
'
Z = √5 √5
= − ? = 5
= 5 → ( )
45)
( + ) = 12 + ( − )
+ 2 + = 12 + − 2 +

35. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
4 = 12 → = 3
=
Q tQ t V
VP PQ
=
(e) t t V
VP VP
=
.
= 15 → ( )
46)
.
+
.
= 8 → + = 8 --------(a)
De E:
=
"( ) ( ) #
( ' ') ( ' ')
=
( )
" ' ' ( ' ')#"( ' ') ( ' '#
( ) 2B :
=
( )
( ')( ')
=
(8 2 2
)
' '
=
V ' '
' '
E = 64 → (2)
47)
( − ?) + ( − L) + (? − L) = 0
→ X
− ? = 0 → = ?
− L = 0 → = L
? − L = 0 → ? = L
= $
QS S uS
(Q u)S
= $
QS QS QS
(Q Q Q)S
= $
QS
SQS
=
.
e
→ ( )

36. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
48)
( + ) + ( − ) = 4
+ 2 + + − 2 + = 4
2 + 2 = 4 → + = 2
− 2 + = 0 → ( − ) = 0 → = − − − ( )
( ) 2B :
= +
= =
= ( )( )
+ 0 = = 1
= 1 → ( )
49)
+ + = 5 → + + ( − 5) = 0
" + + ( − 5)# = 0 → + + ( − 5) = −2" + ( − 5) +
( − 5)#
→ + + − 10 + 25 = −2( + − 5 + − 5 )
Como: + + =41
41 − 10 + 25 = −2( + + ) + 10( a+b)
66 − 10 = −2( + + ) + 10 (5 − )
66 − 50 − 10 + 10 = −2( + + )
( + + ) = −8 − − − − − − − ( )
( ) 202H -4 0 1 -K -4:
( + + ) = 64
+ + + 2( )( + + ) = 64 ---propiedad

37. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
+ + = 184
184 + 2abc(5) =64
= −12 − − − − − − − − − − − − − ( )
De: + + = 5 → ( + + ) = 125
+ + + 3( + + )( + + ) − 3 = 125
(a) y (b) en E:
+ + + 3(5)(−8) − 3(−12) = 125
+ + = 125 + 125 − 36= 209
= + +
= 209 → ( )
50)
+ + = 1
+ + = 1
" + + ( − 1)# = 0 → + + − 2 + 1 = −2( + ( − 1) +
( − 1))
+ + − 2 + 1 = −2( + − + − )
(1) − 2 + 1 = −2 Y + + − ( + )Z
2 − 2 = −2 ( + + ) + 2(1 − )
2 − 2 − 2 + 2 = −2 ( + + )
+ + = 0 − − − − − − − −( )
( + + ) = 1 → + + + 3( + + )( + + ) − 3 = 1 ---
(b)
( ) 2B ( ):
+ + − 3 = 1 ------------(c)
De:
( + + ) = 0
+ + + 2 ( + + ) = 0

38. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
+ + + 2 (1) = 0
+ + = −2 − − − − − (-)
De:
+ + ( − 1) = 0 → Y + + ( − 1)Z = 0
+ + − 2 + 1 = −2" + ( − 1) + ( − 1)#
+ + = 2 − 1 − 2( + − + − )
+ + = 2 − 1 − 2( + + − ( + ))
+ + = 2 − 1 − 2( + + − (1 − ))
+ + = 2 − 1 + 2 − 2 – 2( + + )
+ + = 1 − 2( + + ) − − − −(2)
(-) 2B (2) ∶
+ + = 1 + 4
=
= = P= P
' ' P' P
=
.
. P P
= 1
= 1 → (-)
51)
+ + = 0 → ( + + ) = 0
+ + = 3
+ = 3 − − − − − − ( )
+ = − → ( + ) =
+ + 2 =

39. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
+ = − 2 − − − − − ( )
+ + =
.
( + + ) − − − ( ) EK4ED2- -
( ), ( ) ? ( ) 2B :
=
( P )( = ( = P=)
' ' P'
=
( P )( = ( P =))
8
( P )
=
( = P)
P
=
V ( P)
P
=
V ( P)
( P)
= 3
= 3 → ( )
52)
+ + = 0;
X
+ = −
+ = −
+ = −
----------------------------(a)
=
P
"( ) ( P P)#( P)
=
P
" ( ) P( )#( P)
=
P
( )( P)( P)
( ) 2B :
=
P
( )( )( P)
=
P
P
= −2
= −2 → ( )
53)
+ + = 1 → ( + + ) = 1

40. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
abc = 0 --------------------(a)
+ + + 2( + + ) = 1 -------------(b)
Reemplazando (a):
+ + + 2(0) = 1
+ + = 1 − − − − − − − − − − − −(1)
( + + ) = 1 → + + + 3( + )( + )( + ) = 1
+ + + 3( + + + )( + ) = 1
= = = 0
+ + + 3 ( + ) = 1
+ + + 3( ) + 3( ) = 1
+ + = 1 − − − − − − − − − − − (2)
(1) ? (2) 2B :
=
2
+ 2
+ 2
−
3
+ 3
+ 3
=
.
−
.
=
V
=
.
V
→ (2)
54)
√
=
+ √
=
+ √
=
= 0
(√
=
+ √
=
+ √
=
) = 0 → + + + 3(√
=
+ √
=
)(√
=
+ √
=
)(√
=
+ √
=
) =0
d
√
=
+ √
=
= −√
=
√
=
+ √
=
= −√
=
√
=
+ √
=
= −√
=
------------------(a)
+ + + 3 Y−√
=
ZY −√
=
ZY −√
=
Z = 0
+ + = 3 √
=
√
=
√
=
= 3 √
=
+ + = 3 √
=
→ ( + + ) = 27 − − − ( )
+ + + 3( + )( + )( + ) = 27

41. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
+ + − 27 = − 3( + )( + )( + ) − − − −( )
( ) 2B :
=
= = P= M P
( )( P)( P)
=
( + )( + )( + )
( + )( + )( + )
= −3
= −3 → ( )
55)
+
.
= 2 → − 2 +
.
= 0 → − 2 + 1 = 0
( − 1) = 0 → = 1 − − − − − ( )
( ) 2B :
= $
' mW
m' W
+
= )
.' 8
8W
8
8' .W
+
= $
. .
. .
+
= √1
+
= 1
= 1 → (2)
56)
+
.
Q
= 1 → : +
.
Q
; = 1
+ 2 +
.
Q
= 1 → +
.
Q
= −1----------(a)
+
.
Q
= 1 → : +
.
Q
; = 1
+
.
Q=
+ 3 .
.
Q
: +
.
Q
; = 1

42. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
+
.
Q=
= 1 − 3 : +
.
Q
;
+
.
Q=
= 1 − 3(1)
+
.
Q=
= −2 − − − −( )
( ) ( ):
( +
.
Q
)( +
.
Q=
) = (−1)(−2)
( + .
.
Q=
+ .
.
Q
+
.
QS
) = 2
+
.
QS
= 2 − : +
.
Q
; = 2 − 1
+
.
QS
= 1
= $ +
.
QS
S
= √1
S
= 1 → ( )
57)
=
(Q )"( )' ( )'#
( Q ) ( Q)
=
(Q )"( ) ( ) #"( ) ( ) #
Q Q Q Q
=
(Q )( )(
( Q ) ( Q )
=
(Q )( )( )
(Q ) (Q )
=
< (Q )( )
( )(Q )
= 8 → ( )

43. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
58)
+ ? + L = 0 → ( + ? + L) = 0
+ ? + L = −2( ? + ?L + L) − − − − − ( )
+ ? + L = 0 → ( + ? + L) = 0
+ ? + L = 3 ?L − − − −( )
( ) ( ) ∶
( + ? + L )( + ? + L ) = −6 ?L( ? + ?L + L)
+ ? + L + ? + L + ? + ? L + L + L ? = −6 ?L( ? + ?L +
L)
+ ? + L + (? + ? ) + ( L + L ) + (? L L ? = −6 ?L( ? + ?L +
L)
+ ? + L + ? ( + ?) + L ( + L) + ? L (? + L) = −6 ?L( ? + ?L + L)
+ ? + L + ? (−L) + L (−?) + ? L (− ) = −6 ?L( ? + ?L + L)
+ ? + L − ? L − ?L − ? L = −6 ?L( ? + ?L + L)
+ ? + L − ? L − ?L − ? L = −6 ? L − 6 ? L − 6 ?L
+ ? + L = −5 ? L − 5 ? L − 5 ?L
+ ? + L = −5 ?L ( ? + ?L + L) − − − − − −( )
( ) ? ( ) 2B :
=
QS S uS
Q u
=
Qu (Q u Qu)
(Q u Qu)
= ?
= ? → (-)

44. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
59)
+ .
= 3 → +
.
Q
= 3
: +
.
Q
; = 9 → + 2 :
.
Q
; +
.
Q
= 9
+ 2 +
.
Q
= 9 → +
.
Q
= 7 − − − − − ( )
+
.
Q
= 3 → ( +
.
Q
) = 27
+
.
Q= + 3 .
.
Q
: +
.
Q
; = 27
+
.
Q= = 27 − 3(3)
+
.
Q=
= 18 − − − − − ( )
Sumando ( ) ? ( ):
E = + + +
= +
.
Q
+ +
.
Q= = 7 + 18
= 25 → ( )
60)
= $
(Q u)
Q u
= $
Q u (Q u Qu)
Q u
= $
Q Q Qu Q Q Qu
Q Q Qu
= $
(Q u Qu)
Q u Qu
= √3

45. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
= √3 → ( )
61)
_( ) = + +
_( ) = ( ) + ( ) + ( )
Y : E( ) = (2 − )(1 − ) (1 + )( + 2)
→ (2 − )(1 − ) (1 + )( + 2) = ( ) + ( ) + ( )
(2 − 2 − + )( + 2 + + 2 ) = ( ) + ( ) + ( )
(2 − 3 + )( + 3 + ) = ( ) + ( ) + ( )
(2 + 6 + 4 − 3 − 9 − 6 + + 3 + 2 ) = ( ) + ( ) +
( )
(-5 + + 4) = V
+ +
V
− 5 + 4 = V
+ +
Igualando coeficientes:
X
= 1
= −5
= 4
= + +
= 1 − 5 + 4
= 0 → ( )
62)
_( ) = " − + + − + + − + #
_( ) = " + − ( + + ) + + ( + + )#

46. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
_( ) = " ( + + ) − ( + + ) + ( + + )#
_( ) = " (1) − (1) + 1# = ( − + 1)
_( + + ) = _(1)
_(1) = (1 − 1 + 1) = 1
P(a+b+c) = 1 → (-)
63)
=
( t 9) ( 9 t)
( )(t 9 )
=
t t9 9 9 t9 t
( )(t 9 )
=
( t 9 ) 9 t
( )(t 9 )
=
(t 9 ) (t 9 )
( )(t 9 )
=
( )(t 9 )
( )(t 9 )
= 1 → ( )
64)
= + − − − −( )
= $:
P
; :
P
− ; :
P
− ; :
P
− ;
= $:
+ +
2
; :
+ + −2
2
; :
+ + −2
2
; :
+ + −2
2
;
=
.
[( + + )( + − )( + − )( + − )

47. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
=
.
["( + ) − #" + ( − )#" − ( − )#
=
.
[( + + 2 − )( − ( − ) )
=
.
[( + ) + 2 − )( − − + 2 )
=
.
[( + ) + 2 − )( + 2 − ( + ))
( ) 2B :
=
.
[( + 2 − )( + 2 − )
=
.
[(2 )(2 ) =
.
. 2 ( )
=
P
→ ( )
65)
+ + = 0 − − − − − −( )
+ = − ; + = −
= v P
+
P
+
P
w "
P P
#
= v
.
P
+
.
P
+
P.P
P
w v P P
w
= :
= = P=
P
; : P P
;
+ + = 0 → ( + + ) = 0
+ + = 3
= :
P
P
; : P P
; = 3 : P P
; − − − −( )
+ + = ( + ) − = −
+ + = ( + ) − = −
= 3 :
P
P
; = 3 :
P ( P)
P( P)
;
= 3 (
P P)
P P)
= 3

48. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
= 3 → ( )
66)
(Q )(Q )
Q P
+
(Q )(Q P)
Q
+
(Q )(Q P)
Q
= 1
(Q ) (Q ) (Q ) (Q P) (Q ) (Q P)
( + )( + )( + )
= 1
(Q ) "(Q ) (Q P) # (Q ) (Q P)
( + )( + )( + )
= 1 ------------(a)
( + ) + ( + ) = 0 2B ( )
(Q ) "U# (Q ) (Q P)
( + )( + )( + )
= 1
( + ) ( + ) = ( + )( + )( + )
( + )( + ) = +
Como E =
Q
(Q )(Q P)
=
(Q )(Q P)
(Q )(Q P)
= 1 → ( )
67)
= "( + ) + ( − ) # − 4"(: ; − x: ; y
= v + 2 + + − 2 + w − 4" − #
= (2 + 2 ) -4"
'
'
− 2 +
'
'
#

49. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
= 4 :
'
'
+ 2 +
'
'; − 4
'
'
+ 8 - 4
'
'
= 4
'
'
+ 8 + 4
'
'
−4
'
'
+ 8 - 4
'
'
= 16 → ( )
68)
+ + = − − − − − −( )
+ = − = ( − 1) = − (1 − )
+ = − = ( − 1) = − (1 − )
+ = − = ( − 1) = − (1 − )
De:
.
:
.
; +
.
:
. P
P
; +
.
P
:
. P
P
; =
= −
.
P
:
.
.
; −
.
:
. P
. P
; −
.
P
:
. P
. P
;
= −
.
P
−
.
−
.
P
=
P
P
= −
P
P
=
P
P
= −1 → (2)
69)
? = 1 → ? = 1 − − − − − −( )
=
Q( .)
Q .
+
(Q .)
.
=
Q( .) (Q .)
(Q .)( .)
=
Q( .) (Q Q .)
(Q .)( .)
=
(Q) (Q) Q Q(Q) Q
(Q .)( .)

50. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
( 1 ) en E:
=
(.) (.) Q(.) Q
(Q .)( .)
=
Q Q
(Q .)( .)
=
(Q )
Q Q .
=
(Q )
. Q .
=
(Q )
(Q )
= 2
= 2 → ( )
70)
Y√
'
+ 1ZY√
'
− 1Z(√ + 1)( + 1)( + + 1) =
= Y√
'
− 1ZY√ + 1Z( + 1)( + + 1)
= Y√ − 1Z Y√ + 1Z( + 1)( + + 1)
= ( − 1)( + 1)( + + 1)
= ( − 1)( + + 1)
De:
QW .
Q .
= ( + + 1)
= V
− 1 → ( )
71)
= ( − ? ) − − 2 ? − ?V
= ( − ? ) − ( + 2 ? + ?V
)
= ( − ? ) − ( + ? )

51. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
= "( − ? ) − ( + ? )#" ( − ? ) + ( + ? )#
= " ( − ? )-1][ − ? + 1#
= = ( − ? ) − 1 = + ?V
− 2 ? − 1
= (2) − 2 ? − 1 = 1 − 2 ? − − − − − ( )
Además E:
= ( − ? ) − ( + ? )
E = − 2 ? + ?V
− − 2 ? − ?V
= −4 ? − − − − − − − ( )
(a) =(b)
1 − 2 ? = −4 ? → − 2 ? = 1 --------©
= 1 − 2 ? = 1— 1 = 2
= 2 → ( )
72)
= → − = 0
( − )( + + ) = 0
≠ → + + = 0 → + = − − − − −( )
( ) 2B :
=
( )
=
E = = − = −
.
= −
1
3
→ (-)

52. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
73)
=
( )
+
( P)
P
+
( P )
P
= +
. P eP
P
+
eP V P
P
= − 4 + +
P
− 12 +
eP
+
eP
− 6 +
P
= : +
9
; + :
4
+
9
; + :
4
+ ; − 22
= (
eP
) + :
eP
; + : P
; − 22 − − − −( )
De: a+4b+9c = 0
X
+ 9 = −4
4 + 9 = −
+ 4 = −9
-----------------(b)
( ) 2B :
= − − −
eP
P
− 22 = −4 − 1 − 9 − 22
= −36 → ( )
74)
+ ? + L = 0 → X
+ ? = −L
+ L = −?
? + L = −
− − − − − ( )
( ) 2B :
=
(Q u) (Q u ) ( u Q)
Q u
=
( u u) ( ) ( Q Q)
Q u

53. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
=
eu e eQ
Q u
=
e(Q u )
Q u
= 9
= 9 → ( )
75)
=
"( + ?) + ( − ?) # − 4( − ? )
( + ? ) − ( − ? )
=
(Q Q Q Q ) (Q )
QW Q== W QW Q==W
=
( Q ) (Q )
Q==
=
" Q (Q )#" Q (Q )#
Q==
=
( )( Q )
Q==
=
Q
=
Q
→ ( )
76)
De:
( + + ) = 3( + + )( + + ) − 2( + + ) + 6 −
− − − − − − −EK4ED2- -
( + + ) = 3( + + )(6)-2(-8) +6(2)
( + + ) = 18( + + ) + 16 + 12
( + + ) − 18( + + ) − 28 = 0
Haciendo: + + =
− 18 − 28 = 0;

54. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
(x+2) ( − 2 − 14) = 0
+ 2 = 0 → + + = −2 − − − −( )
+ + = −2 → ( + + ) = −8
+ + + 3( + + )( + + ) − 3 = −8
−8 + 3 (−2)( + + ) − 3(2) = −8
− 6 ( + + ) = 6
+ + = −1 − − − − − ( )
( ) 202H B-4 0 1 4:
( + + ) = −1 →
+ + + 3( + + ) ( + + ) − 3 = −1
Como E = + +
+ 3(−1)Y ( ) + ( ) + ( )Z − 3(2) = −1
− 3Y ( + + )Z − 6 = −1
− 3 "2 (−2)# = 5
= 5 − 12
= −7 → ( )
77)
[ + ? + ? + [ − ? + ? = ? − − − −( )
= [ + ? + ? − [ − ? + ? = ?
( ) E4K :

55. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
? = ([ + ? + ? + [ − ? + ? )([ + ? + ? −
[ − ? + ? )
? = ( + ? + ? ) − ( − ? + ? )
? = + ? + ? − + ? − ?
? = 2 ?
= 2 → (-)
78)
( + 1)( + 1) = ( + ) + 1
+ + + 1 = + 2 + + 1
+ = + +
= ( + ) − ( + ) =( + ) − ( + )
= ( + )(1 − ) − − − − − −( )
+ = + + → = ( + ) − ( + )
= ( + )(1 − ) − − − − − −( )
( ) ? ( ) 2B ;
=
( .)
( .)
=
( .)
( )( .)( .)
=
( )( .)
=
( )( .)
( )( .)
= 1 → ( )
79)
Dividiendo entre sí los datos dados:

56. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
( )( )
( )
=
<<<
U
2+
2
=
.U
+ = :
222
105
; − − − − − − − ( )
= ( ) = ; (a) en E:
=
:
222
105
; +2
:
222
105
; −2
=
'=
8|S
8
8|S
=
.
= 36 → ( )
80)
.
+
.
P
=
P
P
P( )
=
P
→ ( − + )( − + ) = 4 ( − )
( − + ) = 4 ( − )
+ + + 2 − 2 − 2 = 4 − 4
+ + − 2 − 2 + 2 = 0
( + − ) = 0 → + − = 0
= + − − − − − − − ( )
=
P
+
P
+
P
P
; (a) en E:
= +
P
P
+
P
P P
= + 1 +
P
P
= 1 + 1 + 1
= 3 → ( )

57. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
81)
+ + 34 = 10 + 6
( − 10 + 25) + ( − 6 + 9) = 25 + 9 − 34
( − 5) + ( − 3) = 0
}
− 5 = 0
− 3 = 0
→ = 5 ? = 3
E = + + = (5)(3) + 5 + 3 = 15 + 8
= 23 → ( )
82)
+ 2 ? = 1
Sumando 1 a cada lado:
+ 1 = 2 − 2 ? = 2(1 − ?) − − − − − −( )
+ 2 ? = 1 → + 2 ? + ? = 1 + ?
( + ?) = 1 + ? − − − − − ( )
(a) y (b) en E:
= 2 [1 + ? +
(. Q )
[.
=
Q(. ) (. Q )
[.
=
Q Q ( Q)
[.
=
Q Q Q
[.
=
(Q )
[.
=
(Q )
[(Q )
=
(Q )
(P )
= 2
= 2 → ( )

58. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
83)
= + ? − 12
+ ? = 5 → + ? + 2 ? = 25
( + ? ) + 2 ? = 25
11+2xy =25 → ? = 7 − − − −( )
= ( + ? ) − 12
= ( + ?)( − ? + ? ) − 12
(a) Y las hipótesis en E:
= (5)(11 − 7) − 12
= 20 − 12
= 8 → ( )
84)
( + ) + ( − ) = 4
+ 2 + + − 2 + = 4
2 − 4 + 2 = 0 → − 2 + = 0
+( − ) = 0 → =
=
( )=
= =
=
( )=
= =
=
< =
=
= 4 → ( )
85)

59. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
− = − = 2
= − 2
= − 2 − − − − − − − ( )
= + + − − −
= ( + 2) + + ( − 2) − ( + 2) − ( − 2) − ( + 2)( − 2)
= + 4 + 4 + + − 4 + 4 − − 2 − + 2 − + 4
= 12 → (-)
86)
.
Q
+
.
= →
Q
Q
=
+ ? = ?
? = → + ? = − − − − − ( )
De:
( + ?) =
= + 2 ? + ? ; -2 ( ):
= ( + ? ) + 2 ?
= + 2b
= ( + 2) → ( )
87)
+ + = 0 → + = − → ( + ) =
+ 2 + = → − 2 + 2 + 2 + =
− 2 + = − 4

60. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
( − ) = ( − ) = − 4 − − − − − −( )
+ + = 0 → + = − → ( + ) =
+ 2 + = → − 2 + 2 + 2 + =
− 2 + = − 4
( − ) = − 4 − − − − − −( )
+ + = 0 → + = − → ( + ) =
+ 2 + = → − 2 + 2 + 2 + =
− 2 + = − 4
( − ) = − 4 − − − − − −( )
( ), ( ) ? ( ) 2B ;
=
( )
(P )
+
( P)
P( P)
+
(P )
P( P)
=
2
−4
( 2−4 )
+
2
−4
P( 2−4 )
+
2
−4
P( 2
−4 )
=
.
+
.
P
+
.
P
=
P
P
=
P
P
=
U
P
= 0 → ( )
88)
+ + = 0 → ( + + ) = 3 − − − −( )
X
+ = −
+ = −
+ = −
---------------(b)
(a) y (b) en E:
= ( + − ) + ( + − ) + ( + − )
= (− − ) + (− − ) + (− − )

61. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
= (16 ) + 16 + 16
= 16 ( + + )
= 16 :
.
; (3 ) = 16.3 :
.
; (
.
)
= 3 → ( )
89)
.U .U
+ 16 .U .U
= 41
8|
8|
+ 16.
8|
8|
= 41
U
+ 16 U
= 41 .U .U
→ U
+ 8 .U .U
+ 16 U
= 41 .U .U
+ 8 .U .U
U
+ 8 .U .U
+ 16 U
= 49 .U .U
( .U
) + 8 .U .U
+ (4 .U
) = 49 .U .U
( .U
+ 4 .U
) = 49 .U .U
→ .U
+ 4 .U
= √49 .U .U
.U
+ 4 .U
= 7
( ) − 4 + (2 ) = 7 − 4
( − 2 ) = 3 → − 2 = √3
= $
5−2 5
√
S
= $
[3 5 5
√
S
= )$
3 5 5
S
= √ 5 5
8|
= √ → ( )
90)

62. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
De:
( + + ) = + + + 2( + + ) − − − −( )
( + + ) = + + + 3( + + )( + + ) − 3
( + + ) = 3 + 3( + + )( + + ) − 3 − − − −( )
( + + ) = 2 + 2( + + ) − − − − − ( )
(a+b+c) por (c) :
( + + )( + + ) = 2( + + ) + 2( + + )( + + )
( + + ) = 2( + + ) + 2( + + )( + + ) − − − (-)
( ) 2B (-):
3 + 3( + + )( + + ) − 3 = 2( + + ) + 2( + + )( + +
)
3+ ( + + )( + + ) − 3 = 2( + + )
( + + )( + + ) = 3 − 3 + 2( + + ) − − − −(2)
(e) en E:
=
( P)( P P)
. P
=
( P) ( P)( P P)
. P
=
( P) {3 −3+2( + + )}
. P
=
( P) P ( P)
. P
=
P
. P
=
(. P)
( P)
= 3 → (-)
101)
~ =
V9 .
9
=
9
9

63. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
~ =
9
9
~ = BúO2K4 -2 GéKODB4A
~ = 5 → ( )
102)
~ = BúO2K4 -2 GéKODB4A
~ =
9 .
9 <
=
9
9 e
(4B + 12)(B − 9) = (4B − 3)(B − 8)
4B − 24B − 108 = 4B − 35B + 24
11B = 132 → B = 12
~ =
(. ) .
. <
=
VU
= 15 → ( )
103)
• = ~. -2 GéKODB4A
• =
‚
=
‚
E = 1296
E = 36
• =
V
= 12 → ( )

64. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
104)
• = ~. -2 GéKODB4A
• =
9
9 .
=
9 U
9
(3 + 5B)(B + 2) = (5B + 30)(B − 1)
5B + 13B + 6 = 5B + 25B − 30
12n = 36 ; n =3
• =
( )
.
=
.<
• = 9 → ( )
105)
GP = Q
→ • = DOE K
GP = ( u
)ƒ g
( )g .
= Q
2(h − 1) = 24
h = 13
• =
M
u
= 25 → L = 3

65. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
• =
= 25 → ? = 50
L (• − h) = → = 3(25 − 13) = 36
+ ? + L = 3 + 50 + 36
= 89 → (2)
106)
Gg = ( )„ g
(?M
)g .
Gg = 33 → …† = 309
5(~ − 33) + 7(33 − 1) = 309
5~ − 165 + 224 = 309
5 ~ = 250 ; ~ = 50
N = n = 50 → ( )
107)
N =n
Q=(QS)6 (S)6(8|)=
Q6m8 67
=
=
Q=QS6 S6=|
Q6m8 67
=
Q=7S6 =|7S6
Q6m8 67
~ =
9
9 .
=
U 9
9
(3 + 5B)(B + 2) = (30 + 5B)(B − 1)

66. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
5B + 13B + 6 = 5B + 25B − 30
12B = 36 ; B = 3
~ =
9
9 .
=
.
= 9 → (^)
108)
Gg = G
QS S
Q
=
=
( Q)S ( )S
Q
G → −(2 ) . (3?) .
= −(2 ).(3?)
G → −54 ?
Coeficiente → −54 → ( )
109)
G< = ?
V9
9
=
U
= •
• = 10
6n = 10 (n-4) → 6B = 10B − 40
B = 10
QW| '|
QW '

67. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
G< = ( V
).U <
(? )< .
G< = .
? <
→ (-)
110)
GM → …† = 57
GM → ( M
)„ g
(?V
)g .
→ ( M
)„ M
(?V
)M .
→ ( M
)„ M
(?V
)V
→ ( M
)„ M
? V
7(~ − 7) + 36 = 57
7N -49 = 57-36
7N = 70 → ~ = 10
N = N. de términos = 10 → ( )
111)
Qˆ6 6‰
Qˆ ‰
=
(Qˆ)6 (‰)6
Qˆ ‰
t9
t
=
9
‚
= ~
~ = B
G → …† = 42
→ ( t
)„
(?‚
) .
→ O(~ − 5) + 4E = 42 − − − −( )

68. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
G< → …† = 45
→ ( t
)„ <
(?‚
)< .
→ O(~ − 8) + 7E = 45 − − − −( )
El grado relativo a “y” del término 7 es 21:
7p = 21 → E = 3
-2 ( ) ? ( ):
`
OB − 5O + 4E = 42
OB − 8O + 7E = 45
`
OB − 5O + 4(3) = 42
OB − 8O + 7(3) = 45
→ }
OB − 5O = 30
OB − 8O = 24
}
OB − 5O = 30
−OB + 8O = −24
3O = 6 → O = 2
O = 2 → (-)
112)
U.
=
g VU
g
→ BúO2K4 -2 GéKODB4A
4h =
.
. (3h + 60) → 8 h = 3h + 60
5h = 60 → h = 12
h + 1 =
= 12 + 1
= 145 → ( )

69. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
113)
G → .MV V
• = ~. -2 GéKODB4A
• =
'Š
S‹mf
'Š 'Š
S‹mf S‹mf
=
( S‹mf)Œ ( S‹mf)Œ
S‹mf S‹mf
G = Y
‹ e
Z
ƒ
. (
‹ e
) .
Y
‹ e
Z
ƒ
. (
‹ e
) = .MV V
`
(• − 5)(5
− 9) = 176
4(5
− 9) = 64
5
− 9 = 16
16(• − 5) = 176 → • = 16
N = • = 16 → (-)
114)
Gg → …† = 252
Q8W| +|
Q' R
=
(Q')'| (R)'|
Q' R
• = 40 GéKODB4A
Gg
= ( ) U g
(?M
)g .
4(40 − h) + 7(h − 1) = 252

70. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
160 − 4h + 7h − 7 = 252
3 k = 99 → h = 33
Lugar de Gg = 33 → ( )
115)
Como el denominador tiene signo (-) →
20 G2KODB4 16 GD2B2 AD3B4 (−)
G.V = −( ) . .V
. (?).V .
G.V = − .
.(?).
→ (-)
116)
• = ~]
• =
9 .
9 <
=
9
9 e
(4B + 12)(B − 9) = (4B − 3)(B − 8)
4B − 24B − 108 = 4B − 35B + 24
−24B − 108 = −35B + 24
11n = 132
B = 12
• =
(. ) .
. <
= 15
• = 15 → ( )

71. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
117)
Q'ˆ Q'l
Q Qm=
=
(Q ) ˆ (Qm=)m(
'l
=
)
Q Qm=
←
G.U = ( )g .
. ( ) t g
G.U = g
. Vt g
G.U = g Vt g
; k=10
Al ser independiente de n :
U Vt U
= U
20 − 2 − 6O + 30 = 0
O = 8 → 2O = 16
- = 16 → = −12
(Q ) ˆ (Qm=)m(
'l
=
)
Q Qm=
=
(Q )8W (Qm=)8W
Q Qm=
Gg = ( ).V g
. ( )g .
contados desde la derecha
Gg = ( ).V g g
Para que sean racionales enteros -----el numerador ≥ 0
32 − 2h − 3h + 3 ≥ 0

72. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
35 ≥ 5h
5h ≤ 35 → h ≤ 7
N. términos racionales → 7 → ( )
118)
• =
<
t
= 4 → O = 2
Q+ .
Q .
=
(Q )' .
Q .
De:
O + O + O + − − − − − + O9
=
t678 t
t .
Oe
+ O<
+ OM
+ − − − − − − +O + 3 =
=
tf78 t
t .
+ 3
=
t8| t
t .
+ 3 =
t(tf .)
t .
+ 3
=
( f .)
.
+ 3
= 2.U
− 2 + 3
= 2.U
+ 1 → ( )
119)
Gg = GP − − − G2KODB4A DOE K2A

73. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
Qkl Qklml
Qk Qkm8
=
(Qk)l (Qkm8)l
Qk Qkm8
K =
.
GP = ( )
l78
. ( .
)
l78
.
GP = ( )
lm8
. ( .
)
lm8
GP = ( . .
)
lm8
GP = ( .
)
lm8
GP = ( .
)
lm8
GA : ( − 1)(2 − 1) = 30
( − 1)(2 − 1) = 30
a < b:
b -1= par → (2 − 1) 2A DOE K
b-1=2 b= 3 y 2 − 1 = 15 → = 8
b-1 =6 b = 7 y 2a -1 = 5 → = 3
b-1 =10 b= 11 y 2a -1 = 3 → = 2
b-1 = 30 b= 31 y 2a -1 = 1 → = 1
= 7 → ~. -2 GéKODB4A → ( )
120)
(Q )6 6
Q
=
(Q )6 6
(Q )
(Q )6 6
(Q )
= ( + )9 .
+ ( + )9
. + ( + )9
. −
− − − + 9 .
El termino independiente → = 0

74. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
]• = 9 .
+ 9
+ 9
+ − − − + 9 .
]• = 9 .
+ 9 .
+ 9 .
+ − − − − + 9 .
]• = B 9 .
→ (-)
121)
Q=67 S6m8
Q 6mS
=
(Q )
=67
(6mS)
S6m8
6mS
Q 6mS
9
=
9 .
9
3B − 13B − 10 = 10B − 2
3B -23n -8 =0
B =
± √ eV
V
=
±
V
→ b
B. = 8
B = −
.
B = 8
Gg = G.. = ( )
=67
..
. (?9 ).. .
G.. = ( )
=67
..
. (?9 ).U
G.. → ’“” ?
GA = 3B + 2 − 22 + 10B − 50
…† = 13B − 70 = 13(8) − 70
…† = 34 → ( )
122)

75. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
• =
Ut
t .
=
Ut M
t
(20O + 35)(O − 3) = (20O − 57)(O + 1)
20O − 25O − 105 = 20O − 37O − 57
12 O = 48 → O = 4
• =
U( )
.
= 23
• = 23 → ( )
123)
Qkˆ 6
Q Qm=
=
(Q )
kˆ
(Qm=)m
l6
=
Q Qm=
-
9
=
t
→
G.U = ( )g .
. ( )
kˆ
.U
G.U = g U
=kˆ
→ U
2h − 2 + 30 −
t
= 0
2(10) − 2 + 30 −
t
= 0
t
= 48 → O = 32

76. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
B = −
t
= −48
(Q )
kˆ
(Qm=)m
l6
=
Q Qm=
=
(Q )8W (Qm=)8W
Q Qm=
Gg = ( ).V g
( )g .
32-2k +3-3k ≥ 0 → A2 4 GD2B2 G2KODB4A K D4B 02A 2BG2K4A
35 ≥ 5h → 5h ≤ 35
h ≤ 7
Número de términos = 7 → ( )
124)
Qˆ6 6
Qˆ
=
(Qˆ)6 ()6
Qˆ
; n = N. de términos
Gg = ( t
)9 g
. (?)g .
GA = m(n-k)+k-1
G = ( t
)9
. (?) → …† = 21
O (B − 4) + 3 = 21
OB − 4 O = 18 − − − − − − − ( )
Además: considerando el término 2 y el tres:
O(B − 2) + 1 − "O(B − 3) + 2# = 2
OB − 2O + 1 − OB + 3O − 2 = 2
m - 1=2 ; m = 3
De: OB − 4 O = 18
(3)B − 4(3) = 18
B = 10 → ( )

77. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
125)
n =
t
=
‚
M
− − − − − − − (1)
GP = ?MU
→ ~. GéKODB4A DOE K2A
h =
9 .
Gg = ( )9
678
. (?M
)
678
.
Gg = ( )
6m8
. (?M
)
6m8
= ?MU
b
B − 1 =
M
(B − 1) = 70
M
= 70 → = 20
B = + 1 = 21 ; O = 2B = 42 ; E = 147
= E − 3O − 20
= 147 – 3(42) -20
= 1 → (-)
126)

78. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
Qˆ Q6
Q' R
=
(Q')
ˆ
' (QR)
6
R
Q' R
t
=
9
M
Gg” GQ = ( )
ˆ
'
g
(?M
)g .
( )
ˆ
'
g
(?M
)g .
= ‚
? <
----------(a)
( )g .
(?M
)
ˆ
'
g
= .V
? (‚ V)
-----(b)
-2 ( );
}
4O − 16h = 4E
7h − 7 = 28
7h − 7 = 28 → h = 5
O − 20 = E − − − − − ( )
-2 ( ):
`
4h − 4 = 16
7O − 28h = 8E − 48
4 h − 4 = 16 → h = 5
7O − 28(5) = 8E − 48
7O − 8E = 92 ------- (d)
-2 ( ) ? (-);
`
8O − 8E = 160
−7O + 8E = −92
O = 68 ; E = O − 20 = 48
t
=
9
M
→ B = 7 :
t
; = 7(17) = 119

79. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
E = m +n+p
= 68 + 119 + 48
= 235 → (-)
127)
8||
S .
=
( S) | .
S .
† = G.V
G.V” ( ) U .V
. (1).V .
= U
De:
( |)88 ( ')88
| '
, A1 GéKOD 4 2BGK 0 A2Ká:
K =
.. .
= 6
GP = GV
GP = −( U
).. V
. ( )V .
GP = −( U
) . ( ) = − .UU U
→ ( )
128)
• =
9 .
9
=
‚
‚
• = 4 → 5B − 12 = 4B

80. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
B = 12
QS6m8 '‰
Q6 ‰
=
QS(8 )m8 (‰)'
Q8 ‰
=
Q'+ (‰)'
Q8 ‰
(Q8 )' (‰)
Q8 ‰
Gg = ( .
) g
. (?‚
)g .
( .
) g
. (?‚
)g .
= ?
`
48 − 12h = 24
Eh − E = 3
48 − 12h = 24 → 12 h = 24 → h = 2
Eh − E = 3 → E(2 − 1) = 3 → E = 3
= B + E
= 12 + 3 = 15 → ( )
129)
QRS =|
QS
=
(QS)8S ( )8S
QS
Gg = ( )g .
(? ). g
− − − -2 DLT1D2K- -2K2 ℎ
…† -2 h → 5 (h − 1) + 2(15 − h) = 40
5k -5 + 30-2 k = 40
3h = 15 → h = 5
Gg = GM P–9— p– p˜ p˜™˜Pš ›uœ•›˜™p

81. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
GM” ( ). M
(? )M .
GM” ( )<
(? )V
GM = U
?.
GM → …† = 40 + 12
…† = 52 → ( )
130)
Qk7lkl k=7l=7kl
(Q)kl k 7l
=
Qk7lkl k=7l=
.kl
Qklkl k 7l
=
kl(Qk7l k=7l=
)
kl(Qkl ‹k 7l
‹kl )
=
Qk7l k=7l=
Qkl k 7l mkl
• = =
= =
=
( )( )
( )
= +
.
= 1
= 1 → ( )
131)

82. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
GP = 9 → ~] = • = 17
tŠ=m'| 9‹=m88'
tŠ 9‹
=
(tŠ)
Š=m'|
Š (9‹)
‹=m88'
‹
tŠ 9‹
17 =
Q=m'|
Q
=
=m88'
− 17 − 40 = 0 → 1A 4 ž j
(x-5)( + 5 + 8) − 0
→ − 5 = 0 ; = 5
Y :
? − 17? − 114 = 0 → 1A 4 ?ž j
(? − 6)(? + 6? + 19) = 0
→ ? − 6 = 0 ; ? = 6
t+S 98|
tS 9W
=
(tS)8R (9W)8R
tS 9W
GP = Ge

83. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
Ge = (O ).M e
. (BV
)e .
= O U
Bu
(O )<
. (BV
)<
= O U
Bu
48 = z
= + ? + L
= 5 + 6 + 48
= 59 → (-)
132)
N = número de términos = n
(Q )6 6
Q
=
(Q )6 6
(Q )
Penúltimo término → ~ − 1
G„ . = ( + 2?)„ („ .)
. ?„ . .
( + 2?).)
. ?„
= ? + 2?V
(x+2y). ?„
= ? ( + 2?)
→ ~ − 2 = 5
~ = 7 = n → ( )
133)
• =
t
9

84. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
Qˆ7 ˆ7
Q6 6
=
(Q6)Œ (6)Œ
Q6 6
G U = −( 9
)ƒ U
(?9
).e
= − ? <
}
B(• − 20) = 2
19B = 38
B = 2
2 (• − 20) = 2 → • = 21
De: • =
t
9
21 =
t
→ O = 40
= O + B
= 40 + 2 = 42
= 42 → (-)
134)
(Q )=W Q=W
Q
=
(Q )=W Q=W
(Q ) Q
G e = ( + 3) V e
( ) e .
G e = ( + 3)M
( ) <
= −1 → G e = (−1 + 3)M
(−1) <
= 2M (1) = 128
= 128 → ( )

85. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
135)
( ˆ)ˆ78 ( )ˆˆ
ˆ
→
Gg = ( t
)tˆ g( )g .
←
Gg = ( t
)g .( )tˆ g
O (Ot
− h) + h − 1 − "O(h − 1) + Ot
− h# = 4O − 4
O (Ot) − Oh + h − 1 − Oh + O − Ot
+ h = 4O − 4
O. Ot
− Ot
− 2Oh + 2h = 3O − 3
Ot(O − 1) − 2h(O − 1) = 3(O − 1)
Ot
− 2h = 3
h =
tˆ
ht›9 → Ot
− 3 = 0
Si m =3 ; h = 12
ht›9 = 12 → ( )

86. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
136)
• =
t
=
9
= 8
O = 24 ? B = 40
Qˆ 6
Q= S
=
(Q=)+ (S)+
Q= S
G = ( )<
. (? ) .
G = ( ) .(? )
G = e
? U
→ ( )
137)
• =
Ut
t .
=
Ut M
t
(20O + 35)(O − 3) = (O + 1)(20O − 57)
20O − 25O − 105 = 20O − 37 − 57
12 m = 48 → O = 4
• =
U( )
.
• = ~] = 23 → ( )

87. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
138)
• =
9 e
=
9
6B + 18 = 9B ; B = 6
Q=67f =6
Q=
=
Q=(W)7f =(W)
Q=
=
(Q=)f ( )f
Q=
GP =
e .
= 5
GP = G = ( )e
(? ) .
GP = G = ( ) (? )
GP = G = .
?<
= 1 ? ? = 2 → GP = (1). (2)<
GP = 256 → ( )
139)
(Q )8' 8'
Q Q
=
(Q )8' 8'
(Q Q )
=
(Q )8' 8'
(Q )
=
"(Q ) #R ( )R
(Q )
• = ~. -2 GéKODB4A = 7
GP = Gg =
M .
= 4
G = −"( + ) #M
. ( ) .

88. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
G = −"( + ) # . ( )
G = −( + )V V
→ ( )
140)
Q8SS f=
QS =
=
(QS)=8 (=)=8
QS =
• = 31
Gg = ( ) . g
(? )g .
5(31-k) + 3(k-1)= 122
155 − 5h + 3h − 3 = 122 → h = 15
G. = ( ) . .
(? ). .
G. = ( ).V
(? ).
G. = <U
?
DE = diferencia de exponentes:
Ÿ = 80 − 42 = 38 → ( )
141)
Q6 .
Q .
=
G.U = 9 .U
1e
= 9 .U
G U = 9 U
1 e
= 9 U
G.UU = 9 .UU
1ee
= 9 .UU

89. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
G.U. G U. G.UU = V
( 9 .U
). ( 9 U). ( 9 .UU
) = V
9 .VU
= V
3B − 160 = 236
• = B = 132 → ( )
142)
• = 5 =
9
→ B = 15
G = ( ) (1) .
= V
G = ( ) (1) .
= 1
G + G = 1 + V
→ ( )
143)
Recuerde:
Como la diferencia de las X y de “y”de dos posiciones consecutivas es:

90. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
190-195 = - 5
147-140 = 7
El denominador del cociente es: − ?M
Q? ?
QS R
El exponente de las variables x e y será:
( )9
− (?M
)9
(QS)6 (R)6
QS R
Un término k y k+1 será:
Gg = ( )9 g
(?M
)g .
= .e
. ?. U
Gg . = ( )9 (g .)
(?M
)(g .) .
= .eU
. ?. M
}
5B − 5h = 195
7h − 7 = 140
7h − 7 = 140 → h = 21
5B − 5h = 195 → 5B − 5(21) = 195
B = 60 → (-)
144)
GP = œ
? → ~] = • → DOE K
M
‚
=
t
→ EO = 150
(Q‰)
RS
‰ ( )
ˆ
Q‰
GP = ( ‚
)
RS
‰
g
(? )g .
= œ
?
2h − 2 = 24 → h = 13

91. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
GP = G. → • = 25
• =
M
‚
=
t
; E = 3 ? O = 50
‚(M g‚)
‚
= T ; T = 75 − hE → T = 75 − (13)(3)
T = 36
= O + E + T
= 50 + 3 + 36
= 89 → (-)
145)
=6m= =6m=
Q
‰ m8
‰ m8
• =
6
‰ .
=6m= =6m=
Q
‰ m8
‰ m8
=
Q( ‰ m8)Œ ( ‰ m8)Œ
Q
‰ m8
‰ m8
Gg = G = :
‰ .
;
ƒ
. (?
‰ .
).
= .U
?.
b
(• − 2)Y2‚
− 1Z = 210
2‚
− 1 = 15
15(• − 2) = 210 → • = 16
2‚
− 1 = 15 → 2‚
= 2
E = 4 → 4E = 16 − − − − − ( )
• =
6
‰ .
→ 16 =
6
‰ .
=
6
.
39
− 3 = 240 → 39
= 3

92. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
B = 5
De:
= $
‚ 9
= $
.V( )
= 4 → ( )
146)
Qk '
Ql n
=
(Ql)
k
l (n)
'
n
Ql n
Gg = G.< = ( )
k
l
.<
(?P
).< .
( )
k
l
.<
(?P
).M
= ?.M
b :
.<
; = − 54
17 = 17
17 = 17 → = 1
− 18 = − 54 → = 3
Qk '
Ql n
=
Qk '
Q=
= 24 → = 72
= + +
= 72 +3+1
= 76 → ( )

93. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
147)
Qˆ 6
Q =
=
(Q )
ˆ
(=)
6
=
Q =
• =
t
=
9
Gg = GM
= ( )
ˆ
M
(? )M .
= ?
b
:
t
− 7; 2 =
18 =
O − 14 = 18 → O = 32
De:
t
=
9
→ 16 =
9
→ B = 48
(Q )
ˆ
(=)
6
=
Q =
=
(Q )8W (=)8W
Q =
G. = ( ).V .
. (? ). .
G. = ( ).
. (? ).
G. = ?
G. → …† = 2 + 42 = 44 → ( )
148)
Qk=m'| l=m88'
Qk l
GP = Ge → • = ~] = 17

94. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
= U
= 17 =
= ..
→ − 17 − 40 = 0
− 17 − 114 = 0
− 17 − 40 = 0 → ( − 5)( + 5 + 8) = 0
= 5
− 17 − 114 = 0 → ( − 6)( + 6 + 19) = 0
= 6 ž j
Qk=m'| l=m88'
Qk l
=
Q+S 8|
QS W
=
(QS)8R (W)8R
QS W
Ge = ( ).M e
(?V
)<
= U
?P
48 = c
= + +
= 5 + 6 + 48
= 59 → (-)
149)

95. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.
8Sˆ7S| 8Sˆm8|
Qˆ78 ˆm
• =
. t U
t .
=
. t .U
t
(15O + 50)(O − 2) = (15O − 10)(O + 1)
15O + 20O − 100 = 15O + 5O − 10
15m = 90 → O = 6
8Sˆ7S| 8Sˆm8|
Qˆ78 ˆm
=
Q8'| +|
QR '
=
(QR) | (') |
QR '
Gg = ( M
) U g
(? )g .
7(20 − h) + 4(h − 1) = 85
140 – 7k +4k -4 =85
3h = 51 ; h = 17
Gg = 17 → ( )
150)
=| ˆ
Q6
• =
U
9
=
t
= 10
O = 20 ; B = 3
=| ˆ
Q6
=
Q=| |
Q=
=
(Q=)8| ( )8|
Q=
= O + B
= 20 + 3
= 23 → ( )

96. ALGEBRA PREUNIVERSITARIA Ing. Widmar Aguilar, Msc.