Question

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

Answer 1

when f(x) is divided by x-1 it leaves a remainder 5

⇒ 1^4-2.1^3+3.1^2-a.1+b = 5

1-2+3-a+b = 5

-a+b = 3  ------(1)

when f(x) is divided by x+1 it leaves a remainder 19

⇒ (-1)^4-2(-1)^3+3(-1)^2-a(-1) + b = 19

1+2+3+a+b = 19

a+b = 13 -------(2)

solving 1st and 2nd eqns. we get a = 5 and b = 8

In order to determine the remainder of f(x) when it is divided by x-2, substitute the value of x as 2 and the values of a and b in the given polynomial we get

2^4-2.2^3+3.2^2-5.2+8

16-16+12-10+8

= 10 

Hope this helps you.