Let P be the space of polynomials of degree at most k and de

Let P, be the space of polynomials of degree at most k and define the linear map L : P_4 rightarrow P, by Lp := p \"(x) + xp(x). Show that the polynomial q(x) = 1 is not in the range of L.

Solution

if q (x)=1 is in the range then there exist a p(x) = ax^4 + bx^3 + cx^2 + dx + e

such that

p\'\'(x) + xp(x) = 1

comparing coefficients

a =0, b = 0 , c = 0 ,d + 12a =0 ,e + 6b = 0 , 2c = 1

not possible

so no such p exists