Question

☺ please answer the picture with proper solution.

Answer 1

Given that remainder is 2 when divided by x - 1.

f(1) = 2.     -------- (1)

Given that Remainder is 1 when divided by x - 2.

f(2) = 1      --------- (2)


Let ax + b be the remainder and Q be the quotient.

Now,

f(x) = (x - 1)(x - 2)Q(x) + Ax + B    -------------- (*)

Substitute x = 1 in (*), we get

f(1) = 0 + Ax + B

2   = A + B            ----------- (3)


Substitute x = 2 in (*), we get

When x = 2, then

f(2) = 0 + 2A + B

 1 = 2A + B              ----------- (4)


On solving (3) & (4), we get


A + B = 2

2A + B = 1

-----------------

-A = 1

A = -1.


Substitute A = -1 in (3), we get

2 = A + B

2 = -1 + B

B = 3.


Therefore The remainder Ax + B is -x + 3 (or) 3 - x.


Hope this helps!