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Question

# If a polynomial gives remainder 1 and 2 on dividing xâˆ’2 and xâˆ’1 respectively, then the remainder when that polynomial is divided by x2âˆ’3x+2 is-

A
x3
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B
x+3
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C
x2
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D
x+2
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Solution

## The correct option is B −x+3Let we assume that the polynomial isf(x)=(x−1)(x−2)h(x)+px+qNow If we divide f(x) by (x−1) we get remainder 2 that is f(1)=2Because f(x)=(x−1)α(x)+2⇒f(1)=(1−1)α(x)+2=2Sof(1)=(1–1)(1–2)h(1)+p+q=2⇒p+q=2......(1)Similarly, if we divide f(x) by (x−2) we get remainder 1 that is f(2)=1Sof(2)=(2–1)(2–2)h(2)+2p+q=1⇒2p+q=1.....(2)Now subtracting (1) from (2) we get2p+q−(p+q)=1−2Therefore, p=−1andq=2−p=2−(−1)=3Thus, f(x)=(x−1)(x−2)h(x)−x+3So, if we divide f(x) by x2−3x+2=(x−1)(x−2), the remainder R will be −x+3.Hence, the remainder is −x+3.

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