치환으로 나머지

(가), (나)에서 $\left( B \left( 1 \right) \right) ^{4} - 2 \left( B \left( 1 \right) \right) ^{3} - \left( B \left( 1 \right) \right) ^{2} + 2 B \left( 1 \right) = 24$ $\left\{ B \left( 1 \right) - 3 \right\} \left\{ B \left( 1 \right) + 2 \right\} \left\{ \left( B \left( 1 \right) \right) ^{2} - B \left( 1 \right) + 4 \right\} = 0$ $B \left( 1 \right)$은 실수이므로 $B \left( 1 \right) = 3$ 또는 $B \left( 1 \right) = - 2$ $B \left( x \right)$는 최고차항의 계수가 $1$인 일차다항식이므로 $B \left( x \right) = x + 2$ 또는 $B \left( x \right) = x - 3$ (ⅰ) $B \left( x \right) = x + 2$인 경우 $A \left( x \right) = \left\{ B \left( x \right) + 1 \right\} B \left( x \right) \left\{ B \left( x \right) - 1 \right\} \left\{ B \left( x \right) - 2 \right\}$ $A \left( x \right) = \left( x + 3 \right) \left( x + 2 \right) \left( x + 1 \right) x$ $A \left( 2 \right) \neq 0$이므로 조건 (나)를 만족하지 않는다. (ⅱ) $B \left( x \right) = x - 3$인 경우 $A \left( x \right) = \left\{ B \left( x \right) + 1 \right\} B \left( x \right) \left\{ B \left( x \right) - 1 \right\} \left\{ B \left( x \right) - 2 \right\}$ $A \left( x \right) = \left( x - 2 \right) \left( x - 3 \right) \left( x - 4 \right) \left( x - 5 \right)$ $A \left( 2 \right) = 0$이므로 조건 (나)를 만족한다. (ⅰ), (ⅱ)에 의하여 $A \left( x \right)$$= \left( x - 2 \right) \left( x - 3 \right) \left( x - 4 \right) \left( x - 5 \right)$ $= \left\{ \left( x ^{2} - 7 x - 1 \right) + 11 \right\} \left\{ \left( x ^{2} - 7 x - 1 \right) + 13 \right\}$ $= \left( x ^{2} - 7 x - 1 \right) ^{2} + 24 \left( x ^{2} - 7 x - 1 \right) + 143$ 따라서 $A \left( x \right)$를 $x ^{2} - 7 x - 1$로 나눈 나머지는 $143$