p(x)=x^9-2x^3+13x+k what is the remainder of p(x) by (x+1)?

Question

Suppose p(x)=x^9-2x^3+13x+k. The remainder of the division of p(x) by (x-1) is -6. What is the remainder of the division of p(x) by (x-1)?

Answer 1

The remainder after dividing by x-1 is k+12, so k+12=-6 and k=-18.

The remainder after dividing by x+1 is k-12=-30.

The Remainder Theorem tells us that the remainder after dividing by x-1 is p(1)=1-2+13+k=k+12.

Similarly, for x+1 it's p(-1)=-1+2-13+k=k-12.