Álgebra parcial 2
TRANSCRIPT
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RuffiniTiene que ser de grado uno y si es de grado mayor tiene que ser factorable.
DIVISOR
x 2 x 1(x-1)(x-1)
Sea el polinomio
P (x) 4 7 3 12D (x)
x 2
P (x) 4 3 7 2 0 12D (x) x 2(x+2)(x-1)
X=-2 ; x=1
4 -3 7 20 -12
-2 -8 22 -58 76
4 -11 29 -38 64 residuo 1
4 -7 22
4 -7 22 -16 residuo2
Q (x) 4 7 22 Si tuviera dos factoresR (x) = (1mer factor) R2+R1 R1
R (x) = (x+a) R2+R1 R2
R (x) = (x+2) (-16) + 64 R3
R (x) = -16x-22+64 (x+a)R3+(x+b)R2+R1
R (x) = -16x+32
Residuos
(x+a)R3 +(x+b) R2 +R3
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Dividir
P (x) 4 3 7 2 0 12D (x) 2 x 3(x+3) (x+1)
X=-3 ; X=1
4 -3 7 20 -12
-3 -12 45 -156 408
4 -15 52 -136 396
1 4 -11 41
4 -11 41 -95
Q (x) 4 11 41R (x) = (x+a) R2+R1
R (x) = (x+3) (-95) (+396)
R (x) = -95x-285+396
R (x) = -95x+111
Si tuviera 4 residuos
(x+a)R4+(x+b)R3+(x+c)R2+R1
Teorema del resto y del factor
Aplicamos solo si el divisor es de grado 1
D (x)= 1P (x) 2 2 x 1D (x) 3 1 = x= (reemplaso en el polinomio)Resto=? (Residuo)
P (x)
2 2 x 1
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2 13 x 2 13 x 13 1= + = El residuo o resto = R(x)= - Determine si X+2; es divisor del polinomio 5 x = 6P (x) 5 x 6
X=-2
2 52 64-10+6
R(x)=0
(X+2) si es factor del polinomio
Cocientes notablesCociente Es el resultado de una divicin
a b = a b
+ =? No son cocientes notablesx y = x
y = x y= x x y x y x y x y yx y
= x y
= x y
= x x y x y x y x y yDescomposicin factores =(a+b)(a-b)x y = x y m n = m m n m n m n n
m nm n
m nmnmn
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a b = a b (No es posible factorar)a b = a a b b
x
y
= x
y (No es posible factorar)
m n = m m n m n mn nm n = m n (No es posible factorar)TrinomiosTrinomio cuadrado perfecto=
2 x y = 2 x y = 2 1 = 1
Trinomio de la forma a b x c b x c
(a+2)(a+1) (m-3)(m+2)
x 7 x 1 2(x+4)(x+3)
Trinomio de la forma forma normal
= 4m 12 62
= 2 32 22
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= 2 32 12 = 2 3 1
2 m 3= 2 3 1Aspa
-2m
2m 3 3m
m -1 m
= 2 3 1
FRMULA GENERAL
= 4
2
= 1 1 4232.2 = 1 1 2 44 = 1 254
= 1 54 = 12 = 1 54 = 32
19 19a 15
= 4
2
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= 19 36166020 = 1931
20
1 = 1220 = 352 = 193120 2 = 52
1 =35
2 = 52
a 35 = 0 5a 3
a 52 = 0 2a 5
TRINOMIOS INCOMPLETOS
4 4x 4x y y 4
= x
y
4
= x y 2x y 2[= x 2 x y yx 2 x y y= x y x y x -x
2
x
x
x
= x x
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= abx abx= a b x a b x
Descomposicin de polinomios en la formula general
P (x) = 4 7 2 2 2 4 = x 1x 2x 3 x 4
1 4 -7 -22 24
1 1 5 -2 -24
1 5 -2 -24 0 1 factor
2 2 14 24
1 7 12 0
-3 -3 -12
1 4 0
FACTORES (X-1)(X-2)(X+3)(X+4)
P (x) =105 299 108 164x48
105 -299 -108 164 48
-70 246 -92 -48
105 -396 138 72 0
84 -228 -72
105 -285 -90 0
-30 90
105 -315 0
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P (x) =
105
299
108
164x48
= + + 105x315 = , x
= , x = x = , x
23 9 3= 523 32523 32=10 15 3 610 15 3 6
=13 217 9=100 300 225 9 36 36=91 264 189
Casos especiales4x y 4y x x 12x 2x
4x y 4y x x 1 2x= x 4x y 4y 2x 1= x 2y x 1 se factora la diferencia de cuadrados= x 2y x 1x 2y x 1
= x
2y
x
1x
2y
x
1
= x x 2y 1x x 2y 1
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*
25x
20xy4y
12yz9z
30xz
25x 20xy4y 12yz30xz9z5x 2y -3z
5x 2y -3z
10xy -6yz -15xz
10xy -6yz -15xz
20xy -12yz -30xz
(5x+2y-3z)(5x+2y-3z)
(5x+2y-3z)
+ + = =
52 3 62 3 2
5 6 22 32 3 = 5 6 4 62 3 = 5 42 3 = 5 42 3 = 5 42 3
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=
=
=
Fracciones AlgebraicasSi x-y=2n y
+ = 2Calcular el valor de
=+
+
= 2 = 2
= 2
= 2
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. 2 = 2 = 2 2 2
= 2
= = 3
3
= 3.2.2 3 = 6222 322 = 234 12
= 2343 = 12
:
2 3222132 . 6362=
2 3221332=
3122 1 2 = 312221
3 2
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1 12 13 2 a 1
a 1
= 1 12 13 1 2 1 1= 1 12 13 3 2 1 1
1 12 1 4 1= 1 12 1 4
= 1 12 4 1 4= 1 12 8 1 4
1 1
7 4 = 1 4 7 =
7 4 7 =
3 7
RACIONALIZACINab
.
=
FACTOR RACIONALIZANTE23 . = 233
FR = 233233 =2.33
2.3 = 2.36 = 1consiste en racionalizar el numerador o el denominador
1a . = 1
a . 23
23 =
23
33 =
23
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23 = 2. 345
35 . 34 =2345
3 =233
24 = 2. 435
4425 . 435 = 2435
4.4 = 435
8 35xy27xy = 23
23 =3523. 226
3 =428751036
3 22 5 . 22 55 = 2
2. 236 22. 5362 10 10 25 = 2
56 4.12563 = 326 50063
2a .
= 2
.
=
.
=
.
=
=
x2 2 2 = (2 2) 24 . (2 2) 24 (2 2 2 )(2 2) 2 = (2 2) 2
4 42 2 2 = 2 2 2
6 32 . 6 326 32
(2 2 2
) 6 32363.2 = 2 2
6 3230
Ecuaciones lineales
= 19 [ 3 6 5 72 5 ] 13 5 14 = 0
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19 [ 3 6 5 7 1 02 ] 1 3 6 5 141
9[ 3 6 5 3550
2 ] 1 3 6 5 1
4
19 [6 1 2 3 5 5 02 ] 1 3 6 5 146 1 2 3 5 5 018 1 3 6 5 14293818 1 3 6 5 14 58 76 468 2340 9
36 = 4 1 0 2 2 5 5 = 0
= 2255410 = 4512 =5.5
=
2 =
3 2 = 2 3 2 = 2 2 2 = 2 = 2
=( 1 6 )
= 4
1 6 2 1 6 = 1 62 1 6 = 1 6 1 6 2 16 = 2 : 2 16 =
1 6 =
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= 0Ecuaciones de Segundo grado
Resolucin:
FACTORANDO 5 6 = 0X -3=-3x
X -2=-2x
3 2 = 0Completando trinomio cuadrado perfecto 5 6 = 0 5 254 = 6 254 52
= 14
52 = 12 = 12 52 = 3 = 12 52 = 5Formula General
5 6 = 0
= 42 = 5 25 242.1 = 5 12
= 5 12 = 3
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= 5 12 = 2
= = 273 = 9 = 3 3Resolver Ecuacin
= 2 5 2 1 = 5 1 1 = 4 1 = 2 = 1 2
= 1 2
2 1 5 = 1 0 2 152 = 102 5 152 = 0 5 15
2
5 254 = 152 254 52 = 54
= 52 54 = 52 52 = 5 52
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= 52 54 = 5 52
= = 20 4004911636 = 20 4004911636
= 20 377636 = 20 377636 = 2 0 85936 = 2 0 859
36
= 5 259369 = 5 259369
Ecuacin de la forma cuadrtica
= 10 2 4 = 0 10 2 4 = 0 = =
Naturaleza de los radicales
= 42 2 2 = 0
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Discriminante 4 = 1 1 6 = = 17
Si:
D=0 = dos raicis racionales e iguales
D=0 = dos raicis imaginarias diferentes
D=0 = dos raicis reales diferentes
Calcular la naturaleza de las races
= = = 4 9 2 0 72
= 204 = 2 9 72 = 292
7 5 = 0 = 292 72 = 7 292 = 49 202 = 292 72 = 7 292 = 7 292 = 7 29
2
= 7 292
= = 36 36 = 0 3 3 = 0
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= 3 = 3
x
6 x 1 1 = 3 6 4 4
6 364 = 1 1 364 = 8 62
= 84 = 62 2 = 6 222 = 3 2 62 = 2 = 3 2
= 42 42
= 4 42 = 22 =
= 42 42 =
Para el ejercicio anterior
= 3 2 3 2 = 3 2 3 2 = 6 = 3 2 3 2 = 3 2 32 2
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= = ( 3 2 ) 3 2 = 6 = = ( 3 2 ) 3 2 = 9 32 32 2 = 9 2 = 7Encuentre el valor de K:
De tal modo que las dos races sean iguales 2 2 8 = 0 = 0 4 = 0 discriminante22 4.18 = 02 4 4 4 1 6 1 6 3 2 = 04 1 6 1 6 = 0 4 4 = 0 2 2 = 2Ejercicios
La edad del padre es el triple de la de su hijo y dentro de 8 aos ser el doble Cul serla edad de cada uno?
= 3 = 24 8 = 2 8 = 8 3 8 = 2 1 63 2 = 1 6 8
= 8
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Un estudiante se comparte a presentar a su padre la resolucin de 5 problemas
diariamente. El padre dar al hijo $7.50 por cada problema bien resuelto bien resuelto, y
el hijo abona a su padre $6 por cada problema que deje de presentar o este mal resuelto.
Al cabo de 15 das el hijo gan $225 Cuntos problemas resolvi bien el estudiante?
7 . 5 0 6 = 2 2 5 = 7 5 = 7 5 7.504506=22513.5=675 = 5 0