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).
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Answer 1

solution :-

Using Division Algorithm here:-

Dividend=Divisor×Quotient+Remainder

So, Applying it :-

Let q(x),k(x) be quotient when f(x) is divided by x−1 and x−2 respectively

⇒f(x)=(x−1)q(x)+5

∴f(1)=5 ... (1)

Also,f(x)=(x−2)k(x)+7

∴f(2)=7 ... (2)

Now, let ax+b be remainder when f(x) is divided by (x−1)(x−2) and g(x) be quotient.

f(x)=(x−1)(x−2)g(x)+(ax+b)

Using (1) and (2)

5=a+b .... (3)

7=2a+b ....(4)

Solving (3) and (4), we get

a=2 and b=3

∴2x+3 is remainder when f(x) is divided by (x−1)(x−2).

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