Descomposición de una fracción compuesta caso 1

Descomposición de una fracción compuesta
Caso 1
Factores lineales
1)
6
𝑥 2 −3𝑥+3
6
(𝑥−3)(𝑥−1)
=
=
6
(𝑥−3)(𝑥−1)
=
𝐴
𝑥−3
+
𝐵
𝑥−1
(𝑥−1)𝐴+(𝑥−3)𝐵
(𝑥−3)(𝑥−1)
6= (x − 1)A + (x − 3)B
x-3=0
x-1=0
x=3
x=1
6= (3 − 1)A + (3 − 3)B
6= (1 − 1)A + (1 − 3)B
6= 2 A
6= -2 B
A=
6
2
=3
B=
A=3
6
−2
B= -3
6
(𝑥−3)(𝑥−1)
=
3
𝑥−3
+
−3
𝑥−1
https://www.youtube.com/watch?v=FKuNl8pL1Ms&fbclid=IwAR1hDR0WwyfBMbTgKO8mmmJuro84FF8yaHk9ge7OrTETZR4qa90FXFRwQs
2)
5𝑥+1
𝑥 2 −1
5𝑥+1
=
5𝑥+1
(𝑥+1)(𝑥−1)
(𝑥+1)(𝑥−1)
=
=
𝐴
𝑥+1
+
𝐵
𝑥−1
(𝑥−1)𝐴+(𝑥+1)𝐵
(𝑥+1)(𝑥−1)
5x+1 = (x − 1)A + (x + 1)B
X+1=0
x-1=0
x= -1
x= 1
5(-1) +1 = (−1 − 1)A + (−1 + 1)B
5(1) +1 = (1 − 1)A + (1 + 1)B
-5 + 1 = -2 A
5+1 =2B
-4 = -2 A
6 = 2B
−4
6
−2
=A
2
A= 2
5𝑥+1
(𝑥+1)(𝑥−1)
=B
B= 3
=
2
𝑥+1
+
3
𝑥−1
3)
7
x2 +3x−10
7
(𝑥+5)(𝑥−2)
=
=
7
(𝑥+5)(𝑥−2)
=
𝐴
𝑥+5
𝐵
+
𝑥−2
(𝑥−2)𝐴+(𝑥+5)𝐵
(𝑥+5)(𝑥−2)
7= (x − 2)A + (x + 5)B
X+5=0
x-2=0
x=-5
x=2
7= (−5 − 2)A + (−5 + 5)B
7= (2 − 2)A + (2 + 5)B
7= -7 A
7= 7 B
A=
−7
7
= -1
B=
A= -1
7
(𝑥+5)(𝑥−2)
7
7
=1
B= 1
=
−1
𝑥+5
+
1
𝑥−2