A polynomial f(x) with rational coefficients leaves remainder 15, when divided by (x 3),

and remainder (2x + 1), when divided by (x -1)². Find the remainder when f(x) is dividedby (x - 3) . (x - 1)² .

Let The Quotient Obtained Be p(x) And q(x)

Then

f(x) = (x-3)*p(x)+15

Put x = 3 In Above Equation

f(3) = 15

f(x) = (x-1)²*q(x)+2x+1

Put x = 1 In above eq

f(1) = 3

Let The Remainder Be ax+b And quotient obtained be j(x)

f(x) = (x-3)(x-1)²*j(x)+ax+b

put x = 1 and 3 in above equation

f(1) = a+b

f(3) = 3a+b

But We Already Know That f(1)= 3 And f(3) = 15

So

a+b = 3

3a+b = 15

Subtracting We Get

2a = 12

a = 6

And b = -3

So Remainder = 6x-3