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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).
Solution
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).
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