Railways Sample Paper RRB (Assistant Loco Pilot) Sample Test Paper-10

  • question_answer
    The HCF of \[{{X}^{4}}-1\]and\[{{X}^{4}}-2{{X}^{3}}-2{{X}^{2}}-2X-3\]is

    A)  \[({{x}^{2}}+1)\,(x-1)\]     

    B)         \[({{x}^{2}}+1)\]              

    C)         \[({{x}^{2}}+1)\,(x+1)\]   

    D)         \[(x+1)\]                    

    Correct Answer: C

    Solution :

    \[{{x}^{4}}-1=({{x}^{2}}-1)\,({{x}^{2}}+1)=(x-1)\,(x+1)\] \[({{x}^{2}}+1)\] Now \[{{x}^{4}}-2{{x}^{3}}-2{{x}^{2}}-2x-3\] Putting \[x=-1\]in this equation gives 0, so \[(x+1)\] is a factor, divide \[{{x}^{4}}-2{{x}^{3}}-2{{x}^{2}}-2x-3\]by \[(x+1)\] gives \[{{x}^{3}}-3{{x}^{2}}+x-3\] Now put \[x=3,\]gives 0, so another factor is \[(x-3),\] divide \[(x-3)\] gives \[{{x}^{2}}+1\]which cannot be farther divided So\[{{x}^{4}}-2{{x}^{3}}-2{{x}^{2}}-2x-3=({{x}^{2}}+1)(x+1)(x-3)\] Now common factors in both expressions are \[({{x}^{2}}+1)\,(x+1)\] which is the HCF.

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