8.10. Partial Fractions http://www.ck12.org
- 5 x− 2
( 2 x− 1 )( 3 x+ 4 )=
A
2 x− 1 +B
3 x+ 4
5 x− 2 =A( 3 x+ 4 )+B( 2 x− 1 )
5 x− 2 = 3 Ax+ 4 A+ 2 Bx−B
5 = 3 A+ 2 B
[ −^2 =^4 A−B
3 2
4 − 1∣∣
∣∣^5
− 2
]→ · 4 →
→ · 3 →
[ 12 8
12 − 3
∣∣
∣∣^20
− 6
] →
→ −I →
[ 12 8
0 − 11
∣∣
∣∣^20
− 26
]→ · 11 →
→ · 8 →
[ 132 88
0 − 88
∣∣
∣∣^220
− 208
]
→ +II →
→
[ 132 0
0 − 88
∣∣
∣∣−^12208
]→ ÷ 132 →
→ ÷− 88 →
[1 0
0 1
∣∣
∣∣^11261
11]
A= 111 ,B=−^2611
5 x− 2
( 2 x− 1 )( 3 x+ 4 )=111
2 x− 1 +(^2611)
3 x+ 4
2.x+^11 +x+^13 =(x+x 1 +)(^3 x+ 3 )+(x+x 1 )(+^1 x+ 3 )=(x+^21 x)(+x^4 + 3 )
3.
5 x− 2
( 2 x− 1 )( 3 x+ 4 )=
111
2 x− 1 +
(^2611)
3 x+ 4
5 x− 2 = 111 ( 3 x+ 4 )+^2611 ( 2 x− 1 )
55 x− 22 = 3 x+ 4 + 26 ( 2 x− 1 )
55 x− 22 = 3 x+ 4 + 52 x− 26
55 x− 22 = 55 x− 22
Practice
Decompose the following rational expressions. Practice using matrices with at least one of the problems.
1.(x−^31 x)(−x^4 + 4 )
2.x (^22) (xx+−^13 )
3.x(xx+−^15 )
4.x(xx−^2 + 33 )(xx++^16 )
5.^3 xx 22 +(x^2 +x 2 −)^1
6.x(x−x 12 +)(^1 x+ 1 )
7.x^42 (xx^2 −−^94 )
8.(x+^27 x)(−x^4 − 3 )
9.x (^23) (xx 2 −+^41 )
10.(x−^23 )(x+x^52 + 4 )
11.x (^2) (^3 xx−^2 + 3 )(^2 xx− (^2) +^51 )