세 나머지 관계

$A _{1} ( x ) = 2 x ^{3} + 3 x ^{2} + x + 1$, $A _{2} ( x ) = 3 x ^{3} + 8 x ^{2} + 3 x$, $A _{3} ( x ) = 5 x ^{3} + 12 x ^{2} + 3 x - 1$이라 하자. $A _{1} ( x )$, $A _{2} ( x )$, $A _{3} ( x )$를 다항식 $P ( x )$로 나눈 몫을 각각 $Q _{1} ( x )$, $Q _{2} ( x )$, $Q _{3} ( x )$라 하면 $A _{1} ( x ) = P ( x ) Q _{1} ( x ) + R ( x )$ $\cdots \cdots$ ㉠ $A _{2} ( x ) = P ( x ) Q _{2} ( x ) + 2 R ( x )$ $\cdots \cdots$ ㉡ $A _{3} ( x ) = P ( x ) Q _{3} ( x ) + 3 R ( x )$ $\cdots \cdots$ ㉢ ㉠, ㉡에서 $2 A _{1} ( x ) - A _{2} ( x ) = P ( x ) \left\{ 2 Q _{1} ( x ) - Q _{2} ( x ) \right\}$ 즉, $x ^{3} - 2 x ^{2} - x + 2 = P ( x ) \left\{ 2 Q _{1} ( x ) - Q _{2} ( x ) \right\}$이므로 $( x + 1 ) ( x - 1 ) ( x - 2 ) = P ( x ) \left\{ 2 Q _{1} ( x ) - Q _{2} ( x ) \right\}$ $\cdots \cdots$ ㉣ ㉠, ㉢에서 $3 A _{1} ( x ) - A _{3} ( x ) = P ( x ) \left\{ 3 Q _{1} ( x ) - Q _{3} ( x ) \right\}$ 즉, $x ^{3} - 3 x ^{2} + 4 = P ( x ) \left\{ 3 Q _{1} ( x ) - Q _{3} ( x ) \right\}$이므로 $( x - 2 ) ^{2} ( x + 1 ) = P ( x ) \left\{ 3 Q _{1} ( x ) - Q _{3} ( x ) \right\}$ $\cdots \cdots$ ㉤ 이차다항식 $P ( x )$의 최고차항의 계수가 $1$이므로 ㉣, ㉤에서 $P ( x ) = ( x + 1 ) ( x - 2 )$ 따라서 $P ( 3 ) = 4$ [다른 풀이] $A _{1} ( x ) = 2 x ^{3} + 3 x ^{2} + x + 1$, $A _{2} ( x ) = 3 x ^{3} + 8 x ^{2} + 3 x$, $A _{3} ( x ) = 5 x ^{3} + 12 x ^{2} + 3 x - 1$이라 하자. $A _{1} ( x )$, $A _{2} ( x )$, $A _{3} ( x )$를 다항식 $P ( x )$로 나눈 몫을 각각 $Q _{1} ( x )$, $Q _{2} ( x )$, $Q _{3} ( x )$라 하면 $A _{1} ( x ) = P ( x ) Q _{1} ( x ) + R ( x )$ $\cdots \cdots$ ㉠ $A _{2} ( x ) = P ( x ) Q _{2} ( x ) + 2 R ( x )$ $\cdots \cdots$ ㉡ $A _{3} ( x ) = P ( x ) Q _{3} ( x ) + 3 R ( x )$ $\cdots \cdots$ ㉢ ㉠, ㉡, ㉢에서 $A _{3} ( x ) - A _{1} ( x ) - A _{2} ( x )$ $= P ( x ) \left\{ Q _{3} ( x ) - Q _{1} ( x ) - Q _{2} ( x ) \right\} = x ^{2} - x - 2$ 이차다항식 $P ( x )$의 최고차항의 계수가 $1$이므로 $P ( x ) = x ^{2} - x - 2$ 따라서 $P ( 3 ) = 4$