Question

Efectuati calculele :
a) x+1/x-1 + 4/ x²-1 + 1-x/x+1
b) 2x+1/x-2 + 1-2x/x+2 + x²+16/x²-4
c) x-2/x+1 + x+3/x+2 + 5-x² / x²+3x+2
d) x+2/x²-3x+2 + x+1/x²-4x+3 + x³-6x²+11x-6

Answer 1

a) (x+1)/(x-1) + 4/( x²-1) + (1-x)/(x+1)=
=[(x+1)² + 4+(1-x)(x-1)]/(x-1)(x+1) =
=(x²+2x+1+4+x-x²-1+x)/(x-1)(x+1) =
=(4x+4)/(x-1)(x+1) =
=4(x+1)/(x-1)(x+1) =
=4/(x-1)

b) (2x+1)/(x-2) + (1-2x)/(x+2) + (x²+16)/(x²-4)=
= (2x+1)/(x-2) + (1-2x)/(x+2) + (x²+16)/(x-2)(x+2)=
=[(2x+1)(x+2) + (1-2x)(x-2) + (x²+16)]/(x-2)(x+2)=
=(2x²+x+4x+2+x-2x²-2+4x+x²+16)/(x-2)(x+2)=
=(10x+x²+16)/(x-2)(x+2)=
=(x+2)(x+8)/(x-2)(x+2)=
=(x+8)/(x-2)

c) (x-2)/(x+1) + (x+3)/(x+2) + (5-x²) /( x²+3x+2)=
= (x-2)/(x+1) + (x+3)/(x+2) + (5-x²) /( x+1)(x+2)=
=[ (x-2)(x+2)+ (x+3)(x+1) + (5-x²)] /( x+1)(x+2)=
=(x²-4+ x²+3x+x+3 + 5-x²) /( x+1)(x+2)=
=(x²+4x+4) /( x+1)(x+2)=
=(x+2)² /( x+1)(x+2)=
=(x+2) /( x+1)


d) (x+2)/(x²-3x+2) + (x+1)/(x²-4x+3) + (x³-5x²+x+11)/(x³-6x²+11x-6)=
=(x+2)/(x-1)(x-2) + (x+1)/(x-1)(x-3) +(x³-5x²+x+11)/(x-1)(x-2)(x-3)=
=[(x+2)(x-3) +(x-2)(x+1)] /(x-1)(x-2)(x-3)+(x³-5x²+x+11)/(x-1)(x-2)(x-3)=
=(x²+2x-3x-6 +x²-2x+x-2) /(x-1)(x-2)(x-3) +(x³-5x²+x+11)/(x-1)(x-2)(x-3)=
=(2x²-2x-8+x³-5x²+x+11 )/(x-1)(x-2)(x-3) =
=(x³-3x²-x+3) /(x-1)(x-2)(x-3) =
=[x²(x-3)-(x-3)] /(x-1)(x-2)(x-3) =
=(x-3)(x²-1) /(x-1)(x-2)(x-3) =
=(x-3)(x-1)(x+1) /(x-1)(x-2)(x-3) =
=(x+1)/(x-2)