The conversation between human and AI assistant.
How do I show that polynomial [math]p(x)=\frac {p(x+1) + p (x-1)}{2}[/math] has degree atmost 1?
How does this help show that [math]p(x)[/math] has degree at most 1?
Notice that the highest degree term in [math]p(x)[/math] is [math]\frac{1}{2}(x+1)^n[/math], which is an [math]n[/math]-degree polynomial. However, the highest degree term in [math](x-1)^n[/math] is also [math]x^n[/math], which means that [math]p(x)[/math] cannot have a degree greater than 1.