Question

if p(x)=x^4-2x^3+3x^2-ax+b is a polynomial such that when it is divided by x-1 and x+1 the remainders are respectively 5 and 19 .determine the remainder when p(x) is divided by(x-2).find 'a' and 'b' 

Answer 1

remainder = 10
a = 5
b = 8

Answer 2

P(x) = x⁴ - 2 x³ + 3 x² - a x + b

reminder of the polynomial when divided by (x - k)  is  P(k). substitute  k in place of x.

       P(1) = 5  =>  1 -2  +3 - a + b = 5
                b - a = 3

       P(-1)  = 19  =>  1 +2 + 3 + a + b = 19
                 a + b = 13
     adding the two equations we get:    2 b = 16  => b = 8
                 a = 5

   reminder when P(x) is divided by  (x-2) :
   =  P(2)
   = 2^4 - 2 2^3 + 3  2^2 - 5 * 2 + 8
     =  10