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).
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.