Problem
ALG-B3-M07-P001 Additive Polynomial
#1
★★☆☆☆ Level 2 of 5
Find all \(P\in\mathbb R[x]\) such that \(P(x+y)=P(x)+P(y)\) for all \(x,y\).
Compare mixed terms of the highest degree.
With \(y=0\), \(P(0)=0\). If the degree of \(P\) is at least \(2\), then \(P(x+y)\) has a mixed term of highest degree, absent from \(P(x)+P(y)\). Hence \(P\) is linear or zero. Thus \(P(x)=ax+b\), and \(P(0)=0\) gives \(b=0\). The answer is \(P(x)=ax\).
Basic degree-comparison task.