Question

Fie expresia F= n² + 2n + 2 / n² + 2n + 1, unde n ∈ R - {-1}
a) Calculati  valoarea expresiei pentru n= -3
b) Gasiti toate numerele n ∈ N pentru care F ∈ Z
c) Demonstrati ca oricare ar fi n ∈ Q - { -1}, F se poate scrie ca suma patratelor a doua numere rationale

Answer 1

a)F(-3)=(9-6+2)/(9-6+1)=5/4
b)F∈Z
n^2+2n+1|n^2+2n+2
n^2+2n+1|n^2+2n+1
scadem ultimele 2 relatii
n^2+2n+1|1
n^2+2n+1∈{-1,1}
(n+1)^2∈{-1,1}
(n+1)^2≠-1
=> (n+1)^2=1
n+1∈{-1,1}
n∈{-2,0}
c)F=$\frac{n^{2}+2n+2}{n^{2}+2n+1} =\frac{n^{2}+2n+1+1}{n^{2}+2n+1}=\frac{n^{2}+2n+1}{n^{2}+2n+1}+\frac{1}{n^{2}+2n+1}=1+\frac{1}{n^{2}+2n+1}$
1∈Q
$\frac{1}{n^{2}+2n+1}$∈Q