다항식 항등식

[출제의도] 나머지정리를 활용한 추론하기 $\left\{ Q \left( x + 1 \right) \right\} ^{2} + \left\{ Q \left( x \right) \right\} ^{2}$$= \left( x ^{2} - x \right) P \left( x \right)$ $\cdots \cdots$ ㉠ ㉠에 $x$$= 0$, $x$$= 1$을 대입하여 정리하면 $\left\{ Q \left( 1 \right) \right\} ^{2} + \left\{ Q \left( 0 \right) \right\} ^{2}$$= 0$, $\left\{ Q \left( 2 \right) \right\} ^{2} + \left\{ Q \left( 1 \right) \right\} ^{2}$$= 0$ $Q \left( 0 \right)$$= Q \left( 1 \right)$$= Q \left( 2 \right)$$= 0$ 다항식 $Q \left( x \right)$는 최고차항의 계수가 $1$인 삼차다항식이므로 $Q \left( x \right)$$= x \left( x - 1 \right) \left( x - 2 \right)$ $\cdots \cdots$ ㉡ ㉠, ㉡에서 $\left\{ \left( x + 1 \right) x \left( x - 1 \right) \right\} ^{2} + \left\{ x \left( x - 1 \right) \left( x - 2 \right) \right\} ^{2}$ $= x \left( x - 1 \right) P \left( x \right)$ $P \left( x \right)$$= x \left( x - 1 \right) \left\{ \left( x + 1 \right) ^{2} + \left( x - 2 \right) ^{2} \right\}$ $= x \left( x - 1 \right) \left( 2 x ^{2} - 2 x + 5 \right)$ $= x \left( x - 1 \right) \left\{ 2 \left( x - 2 \right) \left( x + 1 \right) + 9 \right\}$ $= 2 x \left( x - 1 \right) \left( x - 2 \right) \left( x + 1 \right) + 9 x \left( x - 1 \right)$ $= 2 \left( x + 1 \right) Q \left( x \right) + 9 x \left( x - 1 \right)$ $R \left( x \right)$$= 9 x \left( x - 1 \right)$ 따라서 $R \left( 3 \right)$$= 9 \times 3 \times 2$$= 54$