Question

A polynomial f(x) when divided by (x - 1) leaves a remainder 3 and when divided by (x - 2) leaves a remainder of 1. Show that when its divided by  (x - i)(x - 2), the remainder is (-2x + 5). 

Answer 1

Given f(x ) =  (x -1 )(x - 2)+(-2x + 5) 

= (x² - 3x + 2) + (-2x + 5)

f(x) = x² - 5x + 7

Substituting = 1 

f(x) = 1 - 5 + 7 =3

when f(x) is divided by (x -1) , remainder = 3

substituting x = 2

f(x) = 4 - 10 + 7 = 1

when f(x) is divided by (x - 2), remainder = 1

`("x"^2 - 5"x" + 7)/("x"^2 - 3"x" + 2) = 1  (-2"x" + 5)/(("x" - 1)("x" - 2))`

and

when f(x) is divided by (x - 1)(x - 2), remainder = (-2x + 5).