Let f(x) be a polynomial such that
f(f(x))−x^2=xf(x).
Find f(−100).
2 answers
101
rearranging,
f(f(x))=x^2 + x*f(x).
assume degree of f(x) to be m.
therefore,f(f(x)) must have degree m^2.
therefore,L.H.S has degree m^2.
let us assume m >2 .
therefore,R.H.S has degree m+1.
now,the polynomial on both sides must have same degree.
hence,m^2=m+1.
but,then m has no integer solution .
therefore our assumption was wrong and m<2.
possible values for m are 0 and 1.
therefore,f(x)=c or f(x)=b*x+c.
now,by trial and error we get f(x)=1-x .
f(f(x))=x^2 + x*f(x).
assume degree of f(x) to be m.
therefore,f(f(x)) must have degree m^2.
therefore,L.H.S has degree m^2.
let us assume m >2 .
therefore,R.H.S has degree m+1.
now,the polynomial on both sides must have same degree.
hence,m^2=m+1.
but,then m has no integer solution .
therefore our assumption was wrong and m<2.
possible values for m are 0 and 1.
therefore,f(x)=c or f(x)=b*x+c.
now,by trial and error we get f(x)=1-x .
1 answer