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Question

If a polynomial gives remainder 1 and 2 on dividing x2 and x1 respectively, then the remainder when that polynomial is divided by x23x+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+3
Let we assume that the polynomial is

f(x)=(x1)(x2)h(x)+px+q

Now If we divide f(x) by (x1) we get remainder 2 that is f(1)=2

Because f(x)=(x1)α(x)+2

f(1)=(11)α(x)+2=2

So

f(1)=(11)(12)h(1)+p+q=2

p+q=2......(1)

Similarly, if we divide f(x) by (x2) we get remainder 1 that is f(2)=1

So

f(2)=(21)(22)h(2)+2p+q=1

2p+q=1.....(2)

Now subtracting (1) from (2) we get

2p+q(p+q)=12

Therefore, p=1

and

q=2p=2(1)=3

Thus, f(x)=(x1)(x2)h(x)x+3

So, if we divide f(x) by x23x+2=(x1)(x2), the remainder R will be x+3.

Hence, the remainder is x+3.

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