MAT03

(,x-l)² -v-l

4    2x^:+2x±\3    = -4r + -&*£- + -%^K.v - 2)(.y² + 1 )² =

(,x-2)(jt^J+l V -^v-²    (.V+1 )²    .V^:+I ¹    '

_ i

2v² + 2x + 13 = /1(.y² + 1 )² + (B.y + C)(.Y - 2) + (D.y + £)(.y³ - 1y² + .y - 2)

3.v+4

(x-l)² x²+l

A’^4	A + D = 0	A = 1
A*^3	-2D + E= 0	B = -3
“> X	2A + B + D-2E=2	^ => C = -4
x'	-2B + C-2D + E = 2	D = -1
X° ,x+2 ,x^2+l ^-	A-2C-2E= 13 J	E = -2

A__j. Bx+C

|.(a-³+1) =

(dla x = 2)

⁵ + l    (X+l )(.X²-.X+1 )    X+1    ,X²-.X+I

a² ± b^} = (a ± />)(rr + ab + b²)

x = A (a^-2 - .y + 1) + (Ry + C)(.y - 1)

_v0

A + B = O -A-B + C = A-C = O

>

d ~ ~j (dla a- = -1)

^B=J

c= 4-

6    —!—

X^J+I

3 .t+1    3 ,x²-x+l

+ S^xt^D Ka-⁴ + I )

x-+j2x+l x²-/2x+\

A-⁴ + 1 = (A-⁴ + 2a-² + 1) - (J2x)² = (a² + J2x + 1) (a-² - J2x + 1)

1 = (Ax + B) (a-² - J2 x + 1) + (Ca- + D) (x² + J2x + I )

,.3

X~

,.0

A + C= O -J2A+B+ J2C + D = O a-J2B + C+ J2D = O B + D = I

"\

A=-±r

2/2

B=i

C = -

2/2

D= ł

4

J2_

4

I.,

1 —* ¹

- .x²-/2.x+l

² .x²+/I.x+l

Zadania

Rozłożyć na ułamki proste następujące funkcje wymierne

'j 5.T*-6.T-t-l

3.

4.

5.

.x³-5.v²+6x '

.r

.x’-3x+2 '

3.x--l l.x

(x-1)^:(.x²+2) ’

,x³+l ’ 1

.x³-l ’

6. ¹

(.x-1)(.x+2)²(.x+3)³

i

,X^J+.X²+I

Opracował: Marian Malec

7.