Polynomial 2

Algebra Level 3

Find the remainder when $x^{100}$ is divided by $x^2-3x+2$ .

( 2-{ 2 }^{ 100 } )

( { 2 }^{ 100 }-1 ) x

(2^{100} - 1) x + (2-2^{100} )

x+( 2-{ 2 }^{ 100 } )

1 solution

Soumya Shrivastva
May 15, 2016

$\Rightarrow { x }^{ 100 }=\left( { x }^{ 2 }-3x+2 \right) \left( Quotient \right) +remainder\\ \quad \quad \quad \quad \quad =\left( { x }^{ 2 }-3x+2 \right) \left( Quotient \right) +ax+b\quad ,\quad where\quad ax+b\quad is\quad the\quad maximum\\ possible\quad value\quad of\quad the\quad remainder.\\ \Rightarrow { x }^{ 100 }=\left( Quo. \right) \left( x-1 \right) \left( x-2 \right) +ax+b\\ \\ For\quad x=1\\ \Rightarrow 1=a+b...................(1)\\ \\ For\quad x=2\\ \Rightarrow { 2 }^{ 100 }=2a+b......................(2)\\ Solving\quad (1)\quad \& \quad (2),\\ a={ 2 }^{ 100 }-1\quad \& \quad b={ 2-2 }^{ 100 }\quad \\ Remainder=ax+b=\left( { 2 }^{ 100 }-1 \right) x+\left( 2-{ 2 }^{ 100 } \right)$

Polynomial 2

1 solution

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