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Question

When a polynomial f(x) is divided by (x-1) the remainder is 5 and when it is divided by (x-2),the remainder is 7.Find the remainder when it is divided by (x-1)(x-2).


Solution

F(x) = (x-1) q(x) + 5 …..1

F(x) = (x-2) q'(x) + 7 ……2

So F(1) = 5 and F(2) = 7.

Now F(x) is divided by (x-1)(x-2) it's is a polynomial of degree two.

Reminder should be a polynomial of degree less then 2. Say.. r=ax+b

Now.. F(x) = (x-1)(x-2) q”(x) + r

F(x) = (x-1)(x-2) q”(x) + (ax+b). ………3

at x=1.
    5= a+b

at x=2.
     7= 2a+b

Solve two equations and get the value of a and b. 
so.
r = 2x+3.


Mathematics

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