Question

When a polynomial p (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 f (x) is divided by (x-1)(x-2)

Answer 1

According to the remainder theorem , if a function is divided by (x-a), remainder obtained is f(a).

Therefore, f(1) = 1 and f(2) = 2 ---------(1)

let the quotient when the polynomial is divided by (x-1)(x-2) be q(x).

As the degree of the divisor is 2, degree of the remainder will be less than 2.

let the remainder be ax+b.

Therefore

f(x) = q(x)(x – 1)(x – 2) + ax + b --------(2)

now put x=1

f(1) = q(x) × 0 + a + b

1 = a + b

i.e, a + b = 1 --------(3)

put x=2

f(2) = q(x) × 0 + (2a + b)

2 = 2a + b

i.e, 2a + b = 2 ----------(4)

From equations (3) and (4), we get

a = 1 and b = 0.

Hence, the remainder is x.