Question

fie fractia algebrica f(x)=4x^3+8x^2-x-2/2x^3+x^2-8x-4 pentru x apartine R-{-2, -1/2; 2}
a)simplificati fractia
b)determinati n apartine Z pentru care F(n) apartine Z​

Answer 1

Explicație pas cu pas:

4x^3+8x^2-x-2=4x^2(x+2)-1(x+2)=(x+2)(4x^2-1)=

(x+2)(2x+1)(2x-1)

2x^3+x^2-8x-4=x^2(2x+1)-4(2x+1)=(2x+1)(x^2-4)=

(2x+1)(x+2)(x-2)

=> F(x)=(x+2)(2x+1)(2x-1)/(2x+1)(x+2)(x-1)=>F(x)=(2x-1)/(x-1)

F(x)€Z <=> (2x-1)/(x-1)€Z <=>x-1|2x-1

x-1|2x-1

x-1|x-1=>x-1|2x-2

=>x-1|2x-1-2x+2=>x-1|1=>x-1€D1 in Z <=>x-1€{-1,1} =>

x€{0,2}

Cum x€R\{-2,-1/2,2} => S€{0}