Question

Polynomial Operators. Define an operator
L:P_n → P_n , L(p(x))= (p(x)-p(1))/(x-1)

a.) Show that L is well-defined, that is, show that the value is in fact a polynomial.

b.) Show that L is linear.

c.) Calculate L(p(x)) where p(x) is one of the polynomials 1, x, x^2,....,x^n
L(1)=
L(x)=
L(x^2)=
.
.
.
L(x^n)=

d.) Give the kernel:
Ker(L)=

e.) Find the image:
Im(L)=