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KVPY Sample Paper KVPY Stream-SX Model Paper-17

  • question_answer
    A certain polynomial \[P(x),\text{ }x~\in R\] when divided by \[x-a,\,x-b,\,x-c\] leaves remainders \[a,\text{ }b,\text{ }c\] respectively. The remainder when P(x) is divided by \[(x-a)(x-b)(x-c)\,\]is (a, b, c are distinct)

    A) 0

    B) x

    C) \[ax+b-c\]

    D) \[a{{x}^{2}}+bx+c\]

    Correct Answer: B

    Solution :

    By remainder theorem, \[P\text{ }\left( a \right)=a,P\left( b \right)=\text{ }b\]and \[P\text{ }\left( c \right)=c.\] Let the required remainder be \[R(x),\] Then \[~P\left( x \right)=\left( x-a \right)\left( x-b \right)\left( x-c \right)Q\left( x \right)+R\left( x \right),\] where \[R(x),\] is a polynomial of degree at most 2. We get \[R\left( a \right)=a,\text{ }R\left( b \right)=b\] and \[R\left( c \right)=\text{ }c.\] So, the equation \[R\left( x \right)-x=\text{ }0\] has three roots \[a,\text{ }b\] and \[c\]. But its degree is at most 2. So, \[R\left( x \right)-x\] must be zero polynomial (or identity). Hence, \[R\left( x \right)=x\]


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