해법
단순화 x+1x100−1
해법
(x2+1)(x48−x46+x44−x42+x40−x38+x36−x34+x32−x30+x28−x26+x24−x22+x20−x18+x16−x14+x12−x10+x8−x6+x4−x2+1)(x24−x23+x22−x21+x20−x19+x18−x17+x16−x15+x14−x13+x12−x11+x10−x9+x8−x7+x6−x5+x4−x3+x2−x+1)(x−1)(x24+x23+x22+x21+x20+x19+x18+x17+x16+x15+x14+x13+x12+x11+x10+x9+x8+x7+x6+x5+x4+x3+x2+x+1)
솔루션 단계
x+1x100−1
지수 규칙 적용
=x+1(x50)2−12
두 제곱 공식의 차이 적용: x2−y2=(x+y)(x−y)(x50)2−12=(x50+1)(x50−1)=x+1(x50+1)(x50−1)
x50+1인수 :(x2+1)(x48−x46+x44−x42+x40−x38+x36−x34+x32−x30+x28−x26+x24−x22+x20−x18+x16−x14+x12−x10+x8−x6+x4−x2+1)
=x+1(x2+1)(x48−x46+x44−x42+x40−x38+x36−x34+x32−x30+x28−x26+x24−x22+x20−x18+x16−x14+x12−x10+x8−x6+x4−x2+1)(x50−1)
지수 규칙 적용
=x+1(x2+1)(x48−x46+x44−x42+x40−x38+x36−x34+x32−x30+x28−x26+x24−x22+x20−x18+x16−x14+x12−x10+x8−x6+x4−x2+1)((x25)2−12)
두 제곱 공식의 차이 적용: x2−y2=(x+y)(x−y)(x25)2−12=(x25+1)(x25−1)=x+1(x2+1)(x48−x46+x44−x42+x40−x38+x36−x34+x32−x30+x28−x26+x24−x22+x20−x18+x16−x14+x12−x10+x8−x6+x4−x2+1)(x25+1)(x25−1)
x25+1인수 :(x+1)(x24−x23+x22−x21+x20−x19+x18−x17+x16−x15+x14−x13+x12−x11+x10−x9+x8−x7+x6−x5+x4−x3+x2−x+1)
=x+1(x2+1)(x48−x46+x44−x42+x40−x38+x36−x34+x32−x30+x28−x26+x24−x22+x20−x18+x16−x14+x12−x10+x8−x6+x4−x2+1)(x+1)(x24−x23+x22−x21+x20−x19+x18−x17+x16−x15+x14−x13+x12−x11+x10−x9+x8−x7+x6−x5+x4−x3+x2−x+1)(x25−1)
x25−1인수 :(x−1)(x24+x23+x22+x21+x20+x19+x18+x17+x16+x15+x14+x13+x12+x11+x10+x9+x8+x7+x6+x5+x4+x3+x2+x+1)
=x+1(x2+1)(x48−x46+x44−x42+x40−x38+x36−x34+x32−x30+x28−x26+x24−x22+x20−x18+x16−x14+x12−x10+x8−x6+x4−x2+1)(x+1)(x24−x23+x22−x21+x20−x19+x18−x17+x16−x15+x14−x13+x12−x11+x10−x9+x8−x7+x6−x5+x4−x3+x2−x+1)(x−1)(x24+x23+x22+x21+x20+x19+x18+x17+x16+x15+x14+x13+x12+x11+x10+x9+x8+x7+x6+x5+x4+x3+x2+x+1)
공통 요인 취소: x+1=(x2+1)(x48−x46+x44−x42+x40−x38+x36−x34+x32−x30+x28−x26+x24−x22+x20−x18+x16−x14+x12−x10+x8−x6+x4−x2+1)(x24−x23+x22−x21+x20−x19+x18−x17+x16−x15+x14−x13+x12−x11+x10−x9+x8−x7+x6−x5+x4−x3+x2−x+1)(x−1)(x24+x23+x22+x21+x20+x19+x18+x17+x16+x15+x14+x13+x12+x11+x10+x9+x8+x7+x6+x5+x4+x3+x2+x+1)