鍗″皵杈捐鍏紡鎺ㄥ

鍗″皵杈捐鍏紡鏄竴涓眰瑙ｄ笁娆℃柟绋嬬殑鍏紡锛岀敱鎰忓ぇ鍒╂暟瀛﹀鍗㈠崱鏂峰崱灏旇揪璇哄湪16涓栫邯棣栧厛鍙戠幇銆傝鍏紡鍙互瑙ｅ喅褰㈠ ax^3 + bx^2 + cx + d = 0 鐨勪笁娆℃柟绋嬶紝骞朵笖鑳藉缁欏嚭瑙ｇ殑绮剧‘鍊笺€?/p>

涓€銆佸紩鍏ヨ櫄鏁?/h3>

鍦ㄦ帹瀵煎崱灏旇揪璇哄叕寮忎箣鍓嶏紝鎴戜滑闇€瑕佸紩鍏ヨ櫄鏁扮殑姒傚康銆傝櫄鏁板彲浠ヨ〃绀轰负 a + bi 鐨勫舰寮忥紝鍏朵腑 a 鍜?b 鍒嗗埆鏄疄鏁伴儴鍒嗗拰铏氭暟閮ㄥ垎锛宨 鏄櫄鏁板崟浣嶏紝婊¤冻 i^2 = -1銆?/p>

閫氳繃寮曞叆铏氭暟锛屾垜浠彲浠ユ墿灞曞疄鏁板煙鍒板鏁板煙锛岃繘鑰岄€氳繃澶嶆暟鏉ヨВ鍐充竴浜涘疄鏁板煙涓嬫棤娉曡В鍐崇殑闂銆?/p>

浜屻€佸皢涓夋鏂圭▼杞寲涓烘爣鍑嗗舰寮?/h3>

涓轰簡鏂逛究鎺ㄥ锛屾垜浠皢涓夋鏂圭▼ ax^3 + bx^2 + cx + d = 0 杞寲涓烘爣鍑嗗舰寮?x^3 + px + q = 0锛屽叾涓?p 鍜?q 鍒嗗埆涓猴細

p = b/a 鍜? q = c/a - (b^2)/(3a^2)

杩欎釜鍙樻崲鍙互閫氳繃浠ｆ崲 y = x + b/(3a) 寰楀埌锛屽嵆锛?/p>

ax^3 + bx^2 + cx + d = 0a(x + b/(3a))^3 + b(x + b/(3a))^2 + c(x + b/(3a)) + d = 0ax^3 + (b/a)x^2 + (c/a - (b^2)/(3a^2))x + (d - (bc)/(3a) + (b^3)/(27a^3)) = 0

浜庢槸鎴戜滑灏卞皢涓夋鏂圭▼杞寲涓轰簡鏍囧噯褰㈠紡銆?/p>

涓夈€佸埄鐢ㄥ鏁版眰鏍?/h3>

鐜板湪鎴戜滑鍙互浣跨敤澶嶆暟鏉ヨВ鍐宠繖涓棶棰樸€傚亣璁句竴缁勫鏁拌В涓?x1, x2 鍜?x3锛屽垯鎴戜滑鏈夛細

x1 + x2 + x3 = 0

x1x2 + x2x3 + x1x3 = p

x1x2x3 = -q

灏嗙涓€涓瓑寮忔敼鍐欎负 x1 = -x2 - x3锛屼唬鍏ョ浜屼釜绛夊紡寰楀埌锛?/p>

(-x2 - x3)x2 + (-x2 - x3)x3 + x2x3 = p

鍖栫畝鍚庡彲寰楋細

x2^2 + x3^2 + x2x3 = -p

灏嗙涓€涓瓑寮忎唬鍏ョ涓変釜绛夊紡骞惰繘琛屽彉褰紝鍒欏彲寰楋細

x1^3 = (x1 + x2 + x3)^3 = -3p x1 - 2q

灏?x1 琛ㄧず涓?a + bi 鐨勫舰寮忥紝鍏朵腑 a 鍜?b 鍒嗗埆鏄疄鏁伴儴鍒嗗拰铏氭暟閮ㄥ垎銆傚垯涓婂紡鍙互琛ㄧず涓猴細

a^3 + 3ab^2a + 3a^2b^2i + b^3(-3b) = -3pa - 3pb^2 + 2q + 3abi(-p + 3b^2)

鎴戜滑鍙互灏嗗疄閮ㄥ拰铏氶儴鍒嗗埆姣旇緝锛屽緱鍒颁袱涓柟绋嬶細

a^3 + 3ab^2a - 3pb^2 = 2q3a^2b^2 - 3b^3 = -3pa

灏嗙浜屼釜鏂圭▼鍐欐垚 a^2 = (-pb)/b^2 - b/3锛屽垯浠ｅ叆绗竴涓柟绋嬪彲寰楋細

a^3 + 3ab^2a - 3pb^2 = 2qa^3 - 3pab^2 - 2qb^2/3= 0

鐜板湪鎴戜滑闇€瑕佹眰瑙ｈ繖涓簩娆℃柟绋嬨€?/p>

鍥涖€佹眰瑙ｄ簩娆℃柟绋?/h3>

鎴戜滑鍙互浣跨敤鏍囧噯鐨勬眰瑙ｄ簩娆℃柟绋嬬殑鏂规硶锛屽嵆姹傚嚭鍏跺垽鍒紡骞跺垎绫昏璁恒€?/p>

璁句簩娆℃柟绋嬩负 y^2 + py + q = 0锛屽垯鍒ゅ埆寮?D = p^2/4 + q^3/27銆傛帴鐫€涓嶅悓鎯呭喌鐨勮璁哄涓嬶細

1. 濡傛灉 D > 0锛屽垯鏈変袱涓疄鏍广€?/p>

灏嗕袱涓疄鏍瑰甫鍏?y^3 = -q/p + y锛屽垯鍙緱鍒颁笁涓鏁拌В x1, x2 鍜?x3銆?/p>

2. 濡傛灉 D = 0锛屽垯鏈変竴涓疄鏍广€?/p>

璁炬鏃剁殑鏍逛负 y1锛屽垯 we write y1 = y2 = 鈭掟潙?, and the third root is y3 = 鈭?饾憹 饾懃1 饾憺3. 锛屽垯鍙緱鍒颁笁涓鏁拌В x1, x2 鍜?x3銆?/p>

3. 濡傛灉 D < 0锛屽垯鏈変竴涓疄鏍瑰拰涓€瀵瑰叡杞櫄鏍广€?/p>

璁惧疄鏍逛负 y1锛屽垯 y2 鍜?y3 鍒嗗埆涓烘牴涓庡叾鍏辫江锛屽嵆 y2 = a - bi 鍜?y3 = a + bi锛屽垯鍙緱鍒颁笁涓鏁拌В x1, x2 鍜?x3銆?/p>

浜斻€佹弿杩板崱灏旇揪璇哄叕寮?/h3>

缁间笂鎵€杩帮紝鎴戜滑寰楀埌浜嗗崱灏旇揪璇哄叕寮忥細

x1 = (-p/3) + u + vx2 = (-p/3) - (u + v)/2 + i(u - v)鈭?/2x3 = (-p/3) - (u + v)/2 - i(u - v)鈭?/2

鍏朵腑 u 鍜?v 鍒嗗埆涓猴細

u = (Q + R)^(1/3)

v = (-Q + R)^(1/3)

Q = (2p^3 - 9pq + 27q^2)/54

R = (p^2 - 3q)/9

閫氳繃杩欎釜鍏紡锛屾垜浠彲浠ユ眰瑙ｄ竴鑸殑涓夋鏂圭▼锛屽苟缁欏嚭鍏剁簿纭€笺€?/p>

鎬荤粨

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