Question

[roseandlisa]

cho đa thức f(x) sao cho:(x^2+x-2).f(x)=f(x+4).chứng minh f(x) có ít nhất 2 nghiệm. 

Answer 1

[Yen Nhi]

\(\left(x^2+x-2\right).f\left(x\right)=f\left(x+4\right)\)

\(\rightarrow\left(x+2x-x-2\right).f\left(x\right)=f\left(x+4\right)\)

\(\rightarrow\left(x-1\right).\left(x+2\right).f\left(x\right)=f\left(x+4\right)\)

\(\text{Với}\)\(x=1\)\(\text{ ta có:}\)\(0.3.f\left(1\right)=f\left(5\right)\)

                                      \(f\left(5\right)=0\)

\(\text{Với}\)\(x=-2\)\(\text{ta có:}\)\(\left(-3\right).0.f\left(-2\right)=f\left(2\right)\)

                                          \(f\left(2\right)=0\)

\(\text{Vậy}\)\(f\left(x\right)\)\(\text{có ít nhất hai nghiệm là:}\)\(x=2\)\(;\)\(x=5\)