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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)

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Solution

## 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.

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