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).

Answer 1

Given when a polynomial f(x) is divided by x – 1 and x – 2, the remainders are 5 and 7 respectively.

So,

We get f(x) = (x - 1) q (x) + 5

We substitute x = 1, we get

f(1) = (1 - 1)q (x) + 5

f(1) = 5

And

f(x) = (x - 2)Q(x) + 7

We substitute x = 2, get

f(2) = 2 (2 - 2) Q (x) + 7

f(2) = 7

Now Let the remainder Ax + B When f(x) divide by (x-1) (x-2), So

f(x) = (x -1) (x -2)p (x) + Ax + B ------(A)

We subtistute x = 1, we

f(1) = (1 - 1) (1 - 2) p (x) + A(1) + B

f(1) = 0 + A + B, Substitute value of f (1) we get

A + B = 5 ----(1)

And we substitute x = 2 in equation A, we get

f(2) = (2 - 1) (2 - 2) p(x) + A(2) + B

f(2) = 0 + 2A + B, Substitute value of f (2) we get

2A + B = 7 ----(2)

Now we substract equation 1 from equation 2, we get

A = 2, Substitute that value in equation 1, we get

2 + B = 5

B = 3

So,

Remainder = Ax + B = (2) x + 3 = 2x + 3