Tarea

1.-
12/(x^2-4)=12/(x^2-2^2 )=12/((X+2)(X-2))=A/((X+2))+B/((X-2))
12=A(X-2)+B(X+2)
12=AX-2A+BX+2B
12=AX+BX-2A+2B
12=(A+B)X+(-2A+2B)
Ec(1): A+B=0
Ec(2): -2A+2B=12
Multiplicar la ecuación (1) por la (2)
2A+2B=0
-2A+2B=12
4B=12
B=12/4
B=3
Sustituyendo (B=3) en Ecuación (1)
A+3=0
A=-3
12/((X+2)(X-2))=(-3)/(X+2)+3/(X-2)12/((X+2)(X-2))=3/(X-2)-3/(X+2)

3.-
(X-1)/(x^2+X)=(X-1)/X(X+1) =A/X+B/(X+1)
X-1=A(X+1)+BX
X-1=AX+A+BX
X-1=(A+B)X+A
A+B=1…….Ec(1)
A=-1……….Ec(2)
Sustituyendo ecuación (2) en ecuación (1)
-1+B=1
B=1+1
B=2
(X-1)/(x^2+X)=(X-1)/(X(X+1))=(-1)/X+2/(X+1)

5.-
(X+5)/(x^2-4X+3)=(X+5)/((X-3)(X-1))=A/(X-3)+B/(X-1)
X+5=A(X-1)+B(X-3)
X+5=AX-A+BX-3B
X+5=(AX+BX+(-A-3B)
X+5=(A+B)X+(-A-3B)
A+B=1………..Ec(1)-A-3B=5……..Ec(2)
SUMAR EC (1) Y EC (2)
A+B=1
-A+B=5
-2B=6
B=6/(-2)=3
Sustituyendo
b=-3 en Ec(1)
A-3=1
A=1+3
A=4
(X+5)/((X-3)=4/(X-3)-3/(X-1)

7.-
(X+12)/(3x^2-5X-2)=(X+12)/((X-2)(3X+1))=A/(X-2)+B/(3X+1)
(3x^2-5X-2)=0
3^2 X^2-3(5X)-6=0
(3x)^2-5(3X)-6=0
(3X-6)(3X+1)=0
3(X-2)(3X+1)=0
(X-2)(3X+1)=0

X+12=A(3X+1)+B(X-2)
X+12=3AX+A+BX-2B
X+12=(3A+B)X+A-2B3A+B=1……Ec(1)
A-2B=12….Ec(2)
Multiplicando Ecuacion 2 por (-3) y Sumarle la Ecuacion (1)
-3(A-2B)=12(-3)
-3A+6B=-36

-3A+6B=-36
3A+B =1
7B=-35=-35/7=-5
Sustituyendo B=-5 en Ec (1)
3A-5=1
3A=1+5
3A=6
A=6/3=2
(X+12)/(X-2)(3X+1) =2/(X-2)-5/(3X+1)

9.-

(3x^2+3X-12)/(6x^3-5x^2-6X)=(3x^2+3X-12)/(X(6x^2+5X-6))=(3x^2+3X-12)/(X(2X+3)(3X-2))=A/X+B/(2X+3)+C/(3X-2)6(6x^2+5X-6)=0
6^2 x^2+6(5X)-36=0
(6X)^2+5(6X)-36=0
(6X+9)(6X-4)=0
3(2X+3)2(3X-2)=0
6(2X+3)(3X-2)=0
(2X+3)(3X-2)=0

〖3x〗^2+3X-12=A(2X+3)(3X-2)+B(X)(3X-2)+C(X)(2X+3)
〖3x〗^2+3X-12=A(6x^2-4X+9X-6)+B(3x^2-2X)+C(2x^2+3X)
〖3x〗^2+3X-12=6Ax^2+5AX-6A+3Bx^2-2BX+2Cx^2+3CX
〖3x〗^2+3X-12=6Ax^2+3Bx^2+2Cx^2+5AX-2BX+3CX-6A
〖3x〗^2+3X-12=(6A+3B+2C) x^2+(5A-2B+3C)X-6A

6A+3B+2C=3
5A-2B+3C=3-6A=-12
Sust.A de ec 3

A=(-12)/(-6)=
A=2

Sust. A=2 en Ec 1 y Ec 2
6(2)+3B+2C=3
12+3B+2C =3
3B+2C =3-12
3B+2C=-9 Ec 4

5(2)-2B+3C=3
10 -2B+3C =3-10
2B+3C=-7 Ec 5

Mult Ec 4 por 2 y
Ec 5 por 3
2(3B+2C)=-9(2)
2B+4C=-18 Ec 6

3(-2B+3C)=-7(3)
-6B+9C=21 Ec 7

Sumar Ec 6 y Ec 76B+4C=-18
-6+9C=-21
13C=-39
C=-39/13=-3
Sustituyendo c=-3 en Ec(4)

3B+2(-3)=-9
3B-6=-9
3B=-9+6
3B=-3
B=-3/3=-1
(3x^2+3X-12)/(X(2X+3)(3X-2))=2/x-1/(2x+3)-3/(3x-2)

11.-

(2x^3)/(x^2-4)

(2x^3)/(x^2-4)=2x+(8x+4)/(x^2-4)

(8x+4)/(x^2-2^2 )=(8x+4)/(x+2)(x-2) =A/(X+2)+B/(X-2)

8X+4=A(X-2)+B(X+2)
8X+4=(AX-2A+BX+2B
8X+4=AX+BX-2A+2B
8X+4=(A+B)X-2A+2B

A+B=8-----(1)
-2A+2B=4---(2)MULTIPLICAR ECUACION 1 POR ECUACION 2

2(A+B)=8(2)
2A+2B=16----EC(3)

SUMAR ECUACION 3 Y ECUACION 2

2A+2B=16
-2A+2B=4
4B=20
B=20/4
B=5

Sustituyendo b=5 en Ec (1)
A+5=8
A=8-5
A=3

(8X+4)/((X+2)(X-2))=3/(X+2)+5/(X-2)+2X

13.-
4X^3-8X^2-10X+30
2X^2+X-6
4X^3-8X^2-10X+30
2X^2+X-5 =2X^2+X-6

7X = 7X =A + B
2X^2+X-6 (X+2)(2X-3) X+2 2X-2
2^2 X^2+2X-12=0
(2X)^2+(2X)-12=0
(2X+4)(2X-3)=0
2(X+2)(2X+3)=0
(X+2)(2X-3)=0

7X=A(2X-3)+B(X+)
7X=2AX-3A+BX+2B
7X=2AX+BX-3A+2B
7X=(2A+B)X-3A+2B

2A+B=7 Ec 1
-3A+2B=0 Ec 2
Mult Ec 1 por (-2)
-2(2A+B)=7(-2)
-4A-2B=-14 Ec 3
Sumar Ec 3 y Ec 2
-4A-2B=-14
-3A+3B=0
-1A =-14A=(-14)/(-7)
A=2

2X-5+ 7 =2X-5+ 2 + 3
2X^2+X-0 X+2 2X-3

15.-
(〖3x〗^2+ 13x-10)/(x^(3 )- 2x^2 ) = (〖3x〗^2+ 13x-10)/(x^(2 ) (x – 2)) = A/X^(2 ) + B/X + C/(X-2)
3x2 + 13x – 10 = A(x-2) Bx(x – 2) + Cx2
3x2 + 13x – 10 = Ax – 2A + Bx2 – 2Bx) + Cx2
3x2 + 13x – 10 = Bx2 + Cx2 + Ax - 2Bx - 2A
3x2 + 13x – 10 = (B + C)x2 + (A –...