Question

find the polynomial of the second degree in x , divisible by x , and such that the remainder of its division by x-1 and x+2 is respectively -2 and 10

Answer 1

Let p(x)=x(ax+b)=ax²+bx (a second degree polynomial degree which is divisible by x).

Using synthetic division on x-1 and x+2:

1 | a     b       0

     a     a | a+b

     a a+b | -2, so a+b=-2, a=-2-b;

-2 | a      b            0

      a   -2a | -2b+4a

      a b-2a |     10, so -2b+4a=10, -2b-8-4b=10, -6b=18, b=-3⇒a=1.

p(x)=x²-3x is the polynomial.

CHECK 

(x²-3x)/(x-1)=x-2-2/(x-1)=(x²-3x+2-2)/(x-1)=(x²-3x)/(x-1)✔️;

(x²-3x)/(x+2)=x-5+10/(x+2)=(x²-3x-10+10)/(x+2)=(x²-3x)/(x+2)✔️.