[{"type":"paragraph","node_id":"llc577jx1wgn","attrs":{},"children":[{"type":"text","text":"最高次数项为3的函数，形如y=ax³+bx²+cx+d（a≠0,b,c,d为"},{"type":"text","text":"常数"},{"type":"text","text":"）的函数叫做三次函数(cubic "},{"type":"text","text":"function"},{"type":"text","text":")。三次函数的图象是一条曲线——回归式"},{"type":"text","text":"抛物线"},{"type":"text","text":"（不同于普通抛物线）。"}]},{"type":"heading","node_id":"1","attrs":{"level":1},"children":[{"type":"text","text":"图象性质"}]},{"type":"paragraph","node_id":"llc577jy1m9a","attrs":{},"children":[{"type":"text","marks":[{"type":"strong"}],"text":"三次函数性态的五个要点"}]},{"type":"paragraph","node_id":"llc577jy8rx","attrs":{},"children":[{"type":"text","text":"⒈三次函数"},{"type":"formula","node_id":"m0fbdy3k1060","attrs":{"formula_height":"2.843","formula_width":"8.672","image_id":"7346097914789249060","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/8e283b641824470889a4ee170626a5a7~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"在"},{"type":"formula","node_id":"m0fbdy3k1n0t","attrs":{"formula_height":"3.343","formula_width":"13.272","image_id":"7346107370830905371","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/821543e785a64fb5aa66285abd2b325b~tplv-xv4ileqgde-image.image","value":"(-∞，+∞)"}},{"type":"text","text":"上的极值点的个数"}]},{"type":"paragraph","node_id":"llc577jy1qha","attrs":{},"children":[{"type":"text","text":"⒉三次函数"},{"type":"formula","node_id":"m0fbdy3kp66","attrs":{"formula_height":"2.843","formula_width":"8.672","image_id":"7346097914789249060","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/8e283b641824470889a4ee170626a5a7~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"的图象与x 轴交点个数"}]},{"type":"paragraph","node_id":"llc577jy16l8","attrs":{},"children":[{"type":"text","text":"⒊单调性问题"}]},{"type":"paragraph","node_id":"llc577jy1uk2","attrs":{},"children":[{"type":"text","text":"⒋三次函数"},{"type":"formula","node_id":"m0fbdy3k21fi","attrs":{"formula_height":"2.843","formula_width":"4.418","image_id":"7346097993462038537","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/13a3400c9fdd43a78353144e0eea1be4~tplv-xv4ileqgde-image.image","value":"f(x)"}},{"type":"text","text":"图象的切线条数"}]},{"type":"paragraph","node_id":"llc577jy10sw","attrs":{},"children":[{"type":"text","text":"⒌融合三次函数和"},{"type":"text","text":"不等式"},{"type":"text","text":"，创设情境求参数的范围"}]},{"type":"block_image","node_id":"llc577jy1030","attrs":{"url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/e2b5b35ef078477ca4ef627b05859f52~tplv-xv4ileqgde-image.image","alt":"三次函数图象和性质A","note":[{"type":"paragraph","node_id":"llc577jy974","attrs":{},"children":[{"type":"text","text":"三次函数图象和性质A"}]}],"align":"center","width":673,"height":438,"id":"7267401069184221239","version":"1","license":"","name":"三次函数图象和性质A","copyright":"","quote_user_id":"","should_transform":false}},{"type":"paragraph","node_id":"llc577jy1pln","attrs":{},"children":[{"type":"text","marks":[{"type":"strong"}],"text":"三次函数的对称中心"}]},{"type":"paragraph","node_id":"llc577jy229q","attrs":{},"children":[{"type":"text","marks":[{"type":"strong"}],"text":"1.三次函数有对称中心的证明"}]},{"type":"paragraph","node_id":"llc577jz217i","attrs":{},"children":[{"type":"text","text":"证明："}]},{"type":"block_image","node_id":"llc577jz66j","attrs":{"url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/f770576a7e364a86ac4ef351a3f80f56~tplv-xv4ileqgde-image.image","alt":"三次函数","note":[{"type":"paragraph","node_id":"llc577jz1gq9","attrs":{},"children":[{"type":"text","text":"三次函数"}]}],"align":"center","width":656,"height":496,"id":"7267400840804040763","version":"1","license":"","name":"三次函数","copyright":"","quote_user_id":"","should_transform":false}},{"type":"paragraph","node_id":"llc577jza7z","attrs":{},"children":[{"type":"text","text":"因为"},{"type":"formula","node_id":"m0fbdy3kgzk","attrs":{"formula_height":"3.176","formula_width":"35.398","image_id":"7346107898193543195","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/5ef52af75a5743c1a0a2f1ce1d4c86c4~tplv-xv4ileqgde-image.image","value":"f(x)=a(x-x_0)^3 +b(x-x_0)+y_0"}},{"type":"text","text":"的对称中心是"},{"type":"formula","node_id":"m0fbdy3k20m","attrs":{"formula_height":"2.843","formula_width":"7.421","image_id":"7346096667063615551","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/587888d185bf4beda17b8239f50954a9~tplv-xv4ileqgde-image.image","value":"(x_{0},y_{0})"}},{"type":"text","text":"，即"},{"type":"formula","node_id":"m0fbdy3kcn","attrs":{"formula_height":"3.176","formula_width":"9.996","image_id":"7346107952526278683","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/2f2fe5ae3c434f65b94b37833958c987~tplv-xv4ileqgde-image.image","value":"(x_{0},F_{(x0)})"}}]},{"type":"paragraph","node_id":"llc577jz1j04","attrs":{},"children":[{"type":"text","text":"所以"},{"type":"formula","node_id":"m0fbdy3k9c0","attrs":{"formula_height":"3.176","formula_width":"26.585","image_id":"7346107944771043338","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/773ab5e1ec974a3fa27430ad59031c26~tplv-xv4ileqgde-image.image","value":"f(x)=ax^3 +bx^2 +cx+d"}},{"type":"text","text":"如果能写成"},{"type":"formula","node_id":"m0fbdy3k210m","attrs":{"formula_height":"3.176","formula_width":"35.398","image_id":"7346107931395555379","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/28a3804eb06f42ef83d4045fd000d12f~tplv-xv4ileqgde-image.image","value":"f(x)=a(x-x_0)^3+b(x-x_0)+y_0"}},{"type":"text","text":"那么三次函数的对称中心就是"},{"type":"formula","node_id":"m0fbdy3k1ao0","attrs":{"formula_height":"3.176","formula_width":"9.996","image_id":"7346107952526278683","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/2f2fe5ae3c434f65b94b37833958c987~tplv-xv4ileqgde-image.image","value":"(x_{0},F_{(x0)})"}},{"type":"text","text":"。"}]},{"type":"paragraph","node_id":"llc577jzohu","attrs":{},"children":[{"type":"text","text":"所以设"},{"type":"formula","node_id":"m0fbdy3k1ebn","attrs":{"formula_height":"3.176","formula_width":"34.085","image_id":"7346107952681533491","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/65089605856a432e8ba76c7e88bfe4ff~tplv-xv4ileqgde-image.image","value":"f(x)=a(x+m)^3 +p(x+m)+n"}}]},{"type":"paragraph","node_id":"llc577jz1hwz","attrs":{},"children":[{"type":"text","text":"得"},{"type":"formula","node_id":"m0fbdy3k11yl","attrs":{"formula_height":"3.176","formula_width":"53.713","image_id":"7346107914706518025","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/13f2c99eb0a445b1bf50445db095de99~tplv-xv4ileqgde-image.image","value":"f(x)=ax^3 +3amx^2 +(3am^2 +p)x+am^3 +pm+n"}}]},{"type":"paragraph","node_id":"llc577jzcrh","attrs":{},"children":[{"type":"text","text":"所以"},{"type":"formula","node_id":"m0fbdy3k177e","attrs":{"formula_height":"3.009","formula_width":"43.77","image_id":"7346107959195271218","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/8c6526273cf84f23a84276371e04073d~tplv-xv4ileqgde-image.image","value":"3am=b; 3am^2 +p=c; am^3 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-\\frac{bc}{3a}"}}]},{"type":"paragraph","node_id":"llc577jz1aes","attrs":{},"children":[{"type":"text","text":"得证。"}]},{"type":"paragraph","node_id":"llc577jz11hu","attrs":{},"children":[{"type":"text","marks":[{"type":"strong"}],"text":"2.推广"}]},{"type":"paragraph","node_id":"llc577jzl44","attrs":{},"children":[{"type":"text","text":"如果f(x）是一个n次多项式，"},{"type":"formula","node_id":"m0fbdy3k1dn2","attrs":{"formula_height":"2.343","formula_width":"5.656","image_id":"7346107955055525926","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/0837c628be5340ac909b1e077a5c1a5b~tplv-xv4ileqgde-image.image","value":"n\\geq 2"}},{"type":"text","text":"（因为直线的对称中心从狭义上讲是没有对称中心而在广义上讲是无数个对称中心），其 "},{"type":"formula","node_id":"m0fbdy3k2263","attrs":{"formula_height":"1.676","formula_width":"1.395","image_id":"7346104256358989859","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/05d7d81dc9674095b5343cf9faef1ae4~tplv-xv4ileqgde-image.image","value":"n"}},{"type":"text","text":"次项系数是"},{"type":"formula","node_id":"m0fbdy3kzjl","attrs":{"formula_height":"2.009","formula_width":"2.284","image_id":"7346107952526409755","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/922dad2cf4574941826343a6f5ebfbc7~tplv-xv4ileqgde-image.image","value":"a_0"}},{"type":"text","text":"，"},{"type":"formula","node_id":"m0fbdy3k1z4k","attrs":{"formula_height":"2.343","formula_width":"5.398","image_id":"7345429163613995058","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/d7ece562fee94309ac3014b7c878c7e3~tplv-xv4ileqgde-image.image","value":"n-1"}},{"type":"text","text":"次项系数是"},{"type":"formula","node_id":"m0fbdy3kjeo","attrs":{"formula_height":"2.009","formula_width":"2.284","image_id":"7346104088767135754","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/e6667657f0a544e2a4779383eeb3e598~tplv-xv4ileqgde-image.image","value":"a_1"}},{"type":"text","text":"，则有"}]},{"type":"block_image","node_id":"m4i4f966196i","attrs":{"url":"//p3-sdbk2-sign.byteimg.com/tos-cn-i-ry4let6mk6/87439601c2e64a3bb0fb4f4acef014ac~tplv-ry4let6mk6-image.image?lk3s=56f7592b&x-expires=1733900960&x-signature=aAHqN%2BXiT106tSKVoHXp1DtTK0o%3D","alt":"三次函数图象A","note":[{"type":"paragraph","node_id":"llc577jz1ua0","attrs":{},"children":[{"type":"text","text":"三次函数图象A"}]}],"align":"center","width":2400,"height":2316,"id":"7446664682850484235","version":"1","license":"","name":"三次函数图象A.jpeg","copyright":"no_copyright","quote_user_id":"","should_transform":false}},{"type":"paragraph","node_id":"llc577jzeol","attrs":{},"children":[{"type":"text","text":"⑴：如果"},{"type":"formula","node_id":"m0fbdy3kvmu","attrs":{"formula_height":"2.843","formula_width":"8.672","image_id":"7346097914789249060","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/8e283b641824470889a4ee170626a5a7~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"的图像是"},{"type":"text","text":"中心对称图形"},{"type":"text","text":"，其对称中心是"},{"type":"formula","node_id":"m0fbdy3k1mhe","attrs":{"formula_height":"5.509","formula_width":"22.196","image_id":"7346107976987476018","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/eef151cad5534c70a293bbef0bb667ab~tplv-xv4ileqgde-image.image","value":"(-\\frac{a_1}{(na_0)}),f(-\\frac{a_1}{(na_0)})"}},{"type":"text","text":";"}]},{"type":"paragraph","node_id":"llc577jz159a","attrs":{},"children":[{"type":"text","text":"⑵：如果"},{"type":"formula","node_id":"m0fbdy3lqvu","attrs":{"formula_height":"2.843","formula_width":"8.672","image_id":"7346097914789249060","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/8e283b641824470889a4ee170626a5a7~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"的图像是轴对称图形，其对称轴是"},{"type":"formula","node_id":"m0fbdy3ln2m","attrs":{"formula_height":"5.509","formula_width":"12.56","image_id":"7346107959195336754","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/cdc5e74f02cd426f9ffb861c8965ad18~tplv-xv4ileqgde-image.image","value":"x=-\\frac{a_1}{(na_0)}"}}]},{"type":"heading","node_id":"2","attrs":{"level":1},"children":[{"type":"text","text":"极值计算"}]},{"type":"paragraph","node_id":"llc577jz1u9r","attrs":{},"children":[{"type":"text","text":"三次函数"},{"type":"formula","node_id":"m0fbdy3lkel","attrs":{"formula_height":"3.176","formula_width":"33.885","image_id":"7346107985405444123","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/e887a1c8f2e745f0987c15c8237062d6~tplv-xv4ileqgde-image.image","value":"f(x)=ax^{3}+bx^{2}+cx+d(a\\neq 0)"}},{"type":"text","text":"，其"},{"type":"internal_link","node_id":"ltwgxr6d1b7l","attrs":{"docId":"385229163457989516","target":1,"title":"导数","doc_id":"385229163457989516"},"children":[{"type":"text","text":"导数"}]},{"type":"text","text":"为"},{"type":"formula","node_id":"m0fbdy3lg8w","attrs":{"formula_height":"3.176","formula_width":"23.196","image_id":"7346107964639559706","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/5ef98fbfeb764a18a5fbe4025ccc7f2e~tplv-xv4ileqgde-image.image","value":"f'(x)=3ax^{2}+2bx+c"}},{"type":"text","text":"。易证当 "},{"type":"formula","node_id":"m0fbdy3lrb6","attrs":{"formula_height":"3.009","formula_width":"9.405","image_id":"7346097956585750540","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/5cef1b218f574458a22c9cbcb7453d23~tplv-xv4ileqgde-image.image","value":"f'(x)=0"}},{"type":"text","text":"有两个不"},{"type":"text","text":"相等"},{"type":"text","text":"的实数根时，f(x)具有极大值和极小值。而当 "},{"type":"formula","node_id":"m0fbdy3l14ue","attrs":{"formula_height":"3.009","formula_width":"9.405","image_id":"7346097956585750540","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/5cef1b218f574458a22c9cbcb7453d23~tplv-xv4ileqgde-image.image","value":"f'(x)=0"}},{"type":"text","text":"有两个相等的实数根或没有实数根时，f(x)不具有极值。"}]},{"type":"paragraph","node_id":"llc577jzdfr","attrs":{},"children":[{"type":"text","text":"若f(x)有极值，设在 "},{"type":"formula","node_id":"m0fbdy3l16zz","attrs":{"formula_height":"2.009","formula_width":"6.812","image_id":"7346107070896242714","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/fee28f9a87d84fdd9c0c7645f9e36329~tplv-xv4ileqgde-image.image","value":"x=x_{1}"}},{"type":"text","text":"和"},{"type":"formula","node_id":"m0fbdy3l16cd","attrs":{"formula_height":"2.009","formula_width":"6.812","image_id":"7346107964639576090","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/9df7aa252eb342afbe51845145b6b9c7~tplv-xv4ileqgde-image.image","value":"x=x_{2}"}},{"type":"text","text":" 处取得，则满足关系式"},{"type":"formula","node_id":"m0fbdy3l1z95","attrs":{"formula_height":"3.009","formula_width":"19.756","image_id":"7346107976987525170","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/120e40dfaf154b10b940f0ee84c87b19~tplv-xv4ileqgde-image.image","value":"f'(x_{1})=f'(x_{2})=0"}},{"type":"text","text":"，因此以下用 "},{"type":"formula","node_id":"m0fbdy3l1ic8","attrs":{"formula_height":"3.176","formula_width":"25.622","image_id":"7346107944771289098","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/7d25ab61256e4e7b96c4da0297fb43fa~tplv-xv4ileqgde-image.image","value":"f(x)=x^{3}+3x^{2}-9x+1"}},{"type":"text","text":"来介绍两种求三次函数极值的方法。"}]},{"type":"heading","node_id":"21","attrs":{"level":2},"children":[{"type":"text","text":"代入原方程法"}]},{"type":"paragraph","node_id":"llc577jz1kr8","attrs":{},"children":[{"type":"text","text":"该方法为"},{"type":"internal_link","node_id":"ltwgxr6d1bx0","attrs":{"docId":"7244797896770732089","target":1,"title":"高中","doc_id":"7244797896770732089"},"children":[{"type":"text","text":"高中"}]},{"type":"text","text":"学生必须掌握的方法，即通过解"},{"type":"text","text":"方程"},{"type":"formula","node_id":"m0fbdy3l1d2b","attrs":{"formula_height":"3.009","formula_width":"9.405","image_id":"7346097956585750540","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/5cef1b218f574458a22c9cbcb7453d23~tplv-xv4ileqgde-image.image","value":"f'(x)=0"}},{"type":"text","text":"，将所得解"},{"type":"formula","node_id":"m0fbdy3lbh4","attrs":{"formula_height":"2.009","formula_width":"2.384","image_id":"7346107985405476891","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/f73b0a1a45f24c8da57f64fbe5f8f317~tplv-xv4ileqgde-image.image","value":"x_1"}},{"type":"text","text":"与"},{"type":"formula","node_id":"m0fbdy3l177k","attrs":{"formula_height":"2.009","formula_width":"2.384","image_id":"7346107952681762867","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/fb17f638e56f430c888546ab7f165d8e~tplv-xv4ileqgde-image.image","value":"x_2"}},{"type":"text","text":"代入f(x)中得到"},{"type":"internal_link","node_id":"ltwgxr6dpn7","attrs":{"docId":"8315160621630271794","target":1,"title":"极值","doc_id":"8315160621630271794"},"children":[{"type":"text","text":"极值"}]},{"type":"text","text":"。"}]},{"type":"paragraph","node_id":"llc577jz2lt","attrs":{},"children":[{"type":"formula","node_id":"m0fbdy3l1fzz","attrs":{"formula_height":"3.176","formula_width":"25.622","image_id":"7346107944771289098","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/7d25ab61256e4e7b96c4da0297fb43fa~tplv-xv4ileqgde-image.image","value":"f(x)=x^{3}+3x^{2}-9x+1"}},{"type":"formula","node_id":"m0fbdy3l1ifz","attrs":{"formula_height":"3.176","formula_width":"57.459","image_id":"7346107985405509659","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/22b53137dd8b4fe5afc2a62048a56e43~tplv-xv4ileqgde-image.image","value":"f'(x)=3x^{2}+6x-9=3(x^{2}+2x-3)=3(x+3)(x-1)"}},{"type":"text","text":"解"},{"type":"formula","node_id":"m0fbdy3l3vd","attrs":{"formula_height":"3.009","formula_width":"9.405","image_id":"7346097956585750540","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/5cef1b218f574458a22c9cbcb7453d23~tplv-xv4ileqgde-image.image","value":"f'(x)=0"}},{"type":"text","text":" 得"},{"type":"formula","node_id":"m0fbdy3l1lz6","attrs":{"formula_height":"2.509","formula_width":"16.132","image_id":"7346107976987607090","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/bba714c54e014224ab31ee8d2cea114b~tplv-xv4ileqgde-image.image","value":"x_{1}=-3,x_{2}=1"}},{"type":"text","text":"。"}]},{"type":"paragraph","node_id":"llc577jz7jj","attrs":{},"children":[{"type":"text","text":"因此极大值："}]},{"type":"paragraph","node_id":"llc577jz96","attrs":{},"children":[{"type":"formula","node_id":"m0fbdy3l1oak","attrs":{"formula_height":"3.176","formula_width":"48.718","image_id":"7346107959195435058","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/d882ea4c1a14452796bd0c69d3712a31~tplv-xv4ileqgde-image.image","value":"f(-3)=(-3)^{3}+3\\times (-3)^{2}-9\\times (-3)+1=28"}},{"type":"text","text":"极小值："},{"type":"formula","node_id":"m0fbdy3lcu4","attrs":{"formula_height":"3.176","formula_width":"36.703","image_id":"7346107952526557211","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/470a38a7639540408c2f38f92e68df9f~tplv-xv4ileqgde-image.image","value":"f(1)=1^{3}+3\\times 1^{2}-9\\times 1+1=-4"}},{"type":"text","text":"该方法简洁明了，但存在一个问题，即如果解出来的"},{"type":"formula","node_id":"m0fbdy3l1yml","attrs":{"formula_height":"2.009","formula_width":"2.384","image_id":"7346107985405476891","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/f73b0a1a45f24c8da57f64fbe5f8f317~tplv-xv4ileqgde-image.image","value":"x_1"}},{"type":"text","text":"与"},{"type":"formula","node_id":"m0fbdy3l1tiv","attrs":{"formula_height":"2.009","formula_width":"2.384","image_id":"7346107952681762867","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/fb17f638e56f430c888546ab7f165d8e~tplv-xv4ileqgde-image.image","value":"x_2"}},{"type":"text","text":"十分复杂（如含有根式，或数字较大等），代入f(x)中计算乘方将是一件不容易的事。下一种方法则可以较好地解决上述问题。"}]},{"type":"heading","node_id":"22","attrs":{"level":2},"children":[{"type":"text","text":"分解因式法"}]},{"type":"paragraph","node_id":"llc577jz154h","attrs":{},"children":[{"type":"text","text":"该方法与方法一有重合的地方，都是先求导，计算"},{"type":"internal_link","node_id":"ltwgxr6d1sjt","attrs":{"docId":"385229163457989516","target":1,"title":"导数","doc_id":"385229163457989516"},"children":[{"type":"text","text":"导数"}]},{"type":"text","text":"零点"},{"type":"formula","node_id":"m0fbdy3lkl0","attrs":{"formula_height":"2.009","formula_width":"2.384","image_id":"7346107985405476891","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/f73b0a1a45f24c8da57f64fbe5f8f317~tplv-xv4ileqgde-image.image","value":"x_1"}},{"type":"text","text":"与"},{"type":"formula","node_id":"m0fbdy3lzp","attrs":{"formula_height":"2.009","formula_width":"2.384","image_id":"7346107952681762867","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/fb17f638e56f430c888546ab7f165d8e~tplv-xv4ileqgde-image.image","value":"x_2"}},{"type":"text","text":"，但接下来的步骤与方法一有区别。"}]},{"type":"paragraph","node_id":"llc577jz41m","attrs":{},"children":[{"type":"text","text":"第二步，计算"},{"type":"formula","node_id":"m0fbdy3l4dw","attrs":{"formula_height":"6.509","formula_width":"5.98","image_id":"7346107976987623474","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/6d1b5249e0bb4d238621bbac9d8d0c2c~tplv-xv4ileqgde-image.image","value":"\\frac {f(x)}{f'(x)}"}},{"type":"text","text":"。因f(x)是三次"},{"type":"internal_link","node_id":"ltwgxr6d1u7","attrs":{"docId":"4089145471017076153","target":1,"title":"多项式","doc_id":"4089145471017076153"},"children":[{"type":"text","text":"多项式"}]},{"type":"text","text":"，而f'(x)是二次多项式，根据多项式的除法运算法则，商一定是一次多项式，而余式也是一次多项式（若余式为"},{"type":"internal_link","node_id":"ltwgxr6dqaf","attrs":{"docId":"315956147310414206","target":1,"title":"常数","doc_id":"315956147310414206"},"children":[{"type":"text","text":"常数"}]},{"type":"text","text":"则无"},{"type":"internal_link","node_id":"ltwgxr6d1mqm","attrs":{"docId":"8315160621630271794","target":1,"title":"极值","doc_id":"8315160621630271794"},"children":[{"type":"text","text":"极值"}]},{"type":"text","text":"）。"}]},{"type":"paragraph","node_id":"llc577jzbqp","attrs":{},"children":[{"type":"text","text":"不妨设"},{"type":"formula","node_id":"m0fbdy3l4qs","attrs":{"formula_height":"3.009","formula_width":"29.929","image_id":"7346108001880703014","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/d4466227e0654113aced9205149378b5~tplv-xv4ileqgde-image.image","value":"f(x)=f'(x)(x-m)+px+q"}},{"type":"text","text":"，因为我们要计算"},{"type":"formula","node_id":"m0fbdy3l1gj9","attrs":{"formula_height":"2.843","formula_width":"5.472","image_id":"7346107353390956584","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/8c35cfa50a1b45859b9e8d47ab60556f~tplv-xv4ileqgde-image.image","value":"f(x_1)"}},{"type":"text","text":"与"},{"type":"formula","node_id":"m0fbdy3l1s5v","attrs":{"formula_height":"2.843","formula_width":"5.472","image_id":"7346107959195484210","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/2552f69a34784abbb420e8709fe372bb~tplv-xv4ileqgde-image.image","value":"f(x_2)"}},{"type":"text","text":"，我们把"},{"type":"formula","node_id":"m0fbdy3l1h5h","attrs":{"formula_height":"2.009","formula_width":"2.384","image_id":"7346107985405476891","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/f73b0a1a45f24c8da57f64fbe5f8f317~tplv-xv4ileqgde-image.image","value":"x_1"}},{"type":"text","text":"与"},{"type":"formula","node_id":"m0fbdy3lpyz","attrs":{"formula_height":"2.009","formula_width":"2.384","image_id":"7346107952681762867","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/fb17f638e56f430c888546ab7f165d8e~tplv-xv4ileqgde-image.image","value":"x_2"}},{"type":"text","text":"代入等号右边的式子中，再考虑到"},{"type":"formula","node_id":"m0fbdy3lf2e","attrs":{"formula_height":"3.009","formula_width":"19.756","image_id":"7346107976987525170","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/120e40dfaf154b10b940f0ee84c87b19~tplv-xv4ileqgde-image.image","value":"f'(x_{1})=f'(x_{2})=0"}},{"type":"text","text":"，立刻可以得到极值"},{"type":"formula","node_id":"m0fbdy3lwac","attrs":{"formula_height":"2.843","formula_width":"33.101","image_id":"7346108003378085926","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/b33d036e885143c88c8f5465069dec67~tplv-xv4ileqgde-image.image","value":"f(x_{1})=px_{1}+q,f(x_{2})=px_{2}+q"}},{"type":"text","text":"。这样一来无论"},{"type":"formula","node_id":"m0fbdy3l1wcl","attrs":{"formula_height":"2.009","formula_width":"2.384","image_id":"7346107985405476891","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/f73b0a1a45f24c8da57f64fbe5f8f317~tplv-xv4ileqgde-image.image","value":"x_1"}},{"type":"text","text":"与"},{"type":"formula","node_id":"m0fbdy3l1ax9","attrs":{"formula_height":"2.009","formula_width":"2.384","image_id":"7346107952681762867","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/fb17f638e56f430c888546ab7f165d8e~tplv-xv4ileqgde-image.image","value":"x_2"}},{"type":"text","text":"有多复杂，都可以避免乘方运算，减少错误。同时我们可以看到，如果"},{"type":"formula","node_id":"m0fbdy3l1oys","attrs":{"formula_height":"6.509","formula_width":"5.98","image_id":"7346107976987623474","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/6d1b5249e0bb4d238621bbac9d8d0c2c~tplv-xv4ileqgde-image.image","value":"\\frac {f(x)}{f'(x)}"}},{"type":"text","text":"的余式仅仅为"},{"type":"text","text":"常数"},{"type":"text","text":"q，那么就有"},{"type":"formula","node_id":"m0fbdy3l160q","attrs":{"formula_height":"2.843","formula_width":"18.21","image_id":"7346107993521586227","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/47f3d50ba8e848669a78b75a52ce7504~tplv-xv4ileqgde-image.image","value":"f(x_{1})=f(x_{2})=q"}},{"type":"text","text":"，因此"},{"type":"internal_link","node_id":"ltwgxr6f1uzy","attrs":{"docId":"8315160621630271794","target":1,"title":"极值","doc_id":"8315160621630271794"},"children":[{"type":"text","text":"极值"}]},{"type":"text","text":"不存在。"}]},{"type":"paragraph","node_id":"llc577jz113r","attrs":{},"children":[{"type":"formula","node_id":"m0fbdy3lni2","attrs":{"formula_height":"3.176","formula_width":"25.622","image_id":"7346107944771289098","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/7d25ab61256e4e7b96c4da0297fb43fa~tplv-xv4ileqgde-image.image","value":"f(x)=x^{3}+3x^{2}-9x+1"}},{"type":"formula","node_id":"m0fbdy3lfh8","attrs":{"formula_height":"3.176","formula_width":"38.914","image_id":"7346108009732390962","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/306f1241ed874acfaa6a2846432476f6~tplv-xv4ileqgde-image.image","value":"f'(x)=3x^{2}+6x-9=3(x^{2}+2x-3)"}},{"type":"text","text":"容易得到等式"},{"type":"formula","node_id":"m0fbdy3l135m","attrs":{"formula_height":"5.176","formula_width":"53.079","image_id":"7346108010256760859","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/8324924c132142deaa08fca5c3693800~tplv-xv4ileqgde-image.image","value":"x^{3}+3x^{2}-9x+1=3(x^{2}+2x-3)\\cdot \\frac {x+1}{3}-8x+4"}},{"type":"text","text":"这里"},{"type":"formula","node_id":"m0fbdy3lc5j","attrs":{"formula_height":"2.509","formula_width":"17.9","image_id":"7346107993521651763","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/6753b58622ff4ccc8d692f8282ad9805~tplv-xv4ileqgde-image.image","value":"px+q=-8x+4"}},{"type":"text","text":"，而解"},{"type":"formula","node_id":"m0fbdy3ljqw","attrs":{"formula_height":"3.009","formula_width":"9.405","image_id":"7346097956585750540","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/5cef1b218f574458a22c9cbcb7453d23~tplv-xv4ileqgde-image.image","value":"f'(x)=0"}},{"type":"text","text":"得"},{"type":"formula","node_id":"m0fbdy3l24q","attrs":{"formula_height":"2.509","formula_width":"16.132","image_id":"7346107976987607090","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/bba714c54e014224ab31ee8d2cea114b~tplv-xv4ileqgde-image.image","value":"x_{1}=-3,x_{2}=1"}},{"type":"text","text":"。"}]},{"type":"paragraph","node_id":"llc577jzhnc","attrs":{},"children":[{"type":"text","text":"代入，得"}]},{"type":"paragraph","node_id":"llc577jz1k7f","attrs":{},"children":[{"type":"formula","node_id":"m0fbdy3l1woe","attrs":{"formula_height":"2.843","formula_width":"52.28","image_id":"7346108026186645513","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/edb5e6fa3f40458384938b28552e0c6b~tplv-xv4ileqgde-image.image","value":"f(-3)=-8\\cdot (-3)+4=28,f(1)=-8\\cdot 1+4=-4"}},{"type":"text","text":"与方法一相比，虽然多了"},{"type":"text","text":"多项式除法"},{"type":"text","text":"这一步，但数字的计算得到了简化。"}]},{"type":"heading","node_id":"3","attrs":{"level":1},"children":[{"type":"text","text":"零点求法"}]},{"type":"heading","node_id":"31","attrs":{"level":2},"children":[{"type":"text","text":"盛金公式法"}]},{"type":"paragraph","node_id":"llc577jz7u9","attrs":{},"children":[{"type":"text","text":"求函数的零点可用盛金公式、"},{"type":"internal_link","node_id":"llc577jz1gj0","attrs":{"docId":"7979614391597994348","target":1,"title":"范盛金","doc_id":"7979614391597994348"},"children":[{"type":"text","text":"范盛金"}]},{"type":"text","text":"判别法或传统解法（卡尔丹公式法）。"}]},{"type":"paragraph","node_id":"llc577jz13xi","attrs":{},"children":[{"type":"text","text":"三次方程"},{"type":"text","text":"应用广泛。用"},{"type":"text","text":"根号"},{"type":"text","text":"解一元三次方程"},{"type":"text","text":"，虽然有著名的卡尔丹公式，并有相应的判别法，但使用卡尔丹公式解题比较复杂，缺乏直观性。我国数学家、高中教师范盛金推导出一套直接用a、b、c、d表达的较简明形式的一元三次方程的一般式新"},{"type":"text","text":"求根公式"},{"type":"text","text":"，并建立了新判别法。"}]},{"type":"paragraph","node_id":"llc577jz1um","attrs":{},"children":[{"type":"text","marks":[{"type":"strong"}],"text":"1.盛金公式"}]},{"type":"paragraph","node_id":"llc577jz1r5b","attrs":{},"children":[{"type":"text","text":"一元三次方程"},{"type":"formula","node_id":"m0fbdy3lqad","attrs":{"formula_height":"2.843","formula_width":"25.314","image_id":"7346108026186661897","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/22436112ce73499cb6efede0fc93a13d~tplv-xv4ileqgde-image.image","value":"aX^3 +bX^2 +cX+d=0"}},{"type":"text","text":"，（"},{"type":"formula","node_id":"m0fbdy3le5e","attrs":{"formula_height":"2.509","formula_width":"12.157","image_id":"7346107985405722651","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/972933a61ab8485b901b64f714f80cc2~tplv-xv4ileqgde-image.image","value":"a,b,c,d∈R"}},{"type":"text","text":"，且"},{"type":"formula","node_id":"m0fbdy3l42y","attrs":{"formula_height":"2.676","formula_width":"5.491","image_id":"7346103572280377396","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/ab5164677f994339875901119539d512~tplv-xv4ileqgde-image.image","value":"a≠0"}},{"type":"text","text":"）"}]},{"type":"paragraph","node_id":"llc577jz10qc","attrs":{},"children":[{"type":"text","text":"重根"},{"type":"text","text":"判别式"}]},{"type":"paragraph","node_id":"llc577jz21tk","attrs":{},"children":[{"type":"text","marks":[{"type":"strong"}],"text":"总判别式"},{"type":"formula","node_id":"m0fbdy3lxa","attrs":{"formula_height":"3.509","formula_width":"14.691","image_id":"7346102492784230435","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/564dc9b3ed7346c1830842b836f5c16a~tplv-xv4ileqgde-image.image","value":"Δ=B^2－4AC"}},{"type":"text","marks":[{"type":"strong"}],"text":"。"}]},{"type":"paragraph","node_id":"llc577jz1yhi","attrs":{},"children":[{"type":"text","marks":[{"type":"strong"}],"text":"当"},{"type":"formula","node_id":"m0fbdy3l17yo","attrs":{"formula_height":"2.176","formula_width":"10.866","image_id":"7346104016075636788","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/45771b831c494b9690283082cbf0d01d~tplv-xv4ileqgde-image.image","value":"A=B=0"}},{"type":"text","marks":[{"type":"strong"}],"text":"时；"}]},{"type":"paragraph","node_id":"llc577jzdq0","attrs":{},"children":[{"type":"text","marks":[{"type":"strong"}],"text":"当"},{"type":"formula","node_id":"m0fbdy3lpbt","attrs":{"formula_height":"3.509","formula_width":"18.952","image_id":"7346108004116398118","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/8cd29ca4aa7b47ba9d6666caf5f16165~tplv-xv4ileqgde-image.image","value":"Δ=B^2 －4AC>0"}},{"type":"text","marks":[{"type":"strong"}],"text":"时；"}]},{"type":"paragraph","node_id":"llc577jz4d2","attrs":{},"children":[{"type":"text","marks":[{"type":"strong"}],"text":"其中，当"},{"type":"formula","node_id":"m0fbdy3ld2s","attrs":{"formula_height":"3.509","formula_width":"18.952","image_id":"7346103932776726528","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/a9f11fa27f264fda9c05906e34e52355~tplv-xv4ileqgde-image.image","value":"Δ=B^2－4AC=0"}},{"type":"text","marks":[{"type":"strong"}],"text":"时；"}]},{"type":"paragraph","node_id":"llc577jz7sz","attrs":{},"children":[{"type":"text","marks":[{"type":"strong"}],"text":"当"},{"type":"formula","node_id":"m0fbdy3l1hyu","attrs":{"formula_height":"3.509","formula_width":"18.952","image_id":"7346108032419364891","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/ac3391ca0ce74ecb9f6ffbbc9a69d885~tplv-xv4ileqgde-image.image","value":"Δ=B^2 －4AC<0"}},{"type":"text","marks":[{"type":"strong"}],"text":"时；（详细见图）"}]},{"type":"block_image","node_id":"llc577jz1n47","attrs":{"url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/f770576a7e364a86ac4ef351a3f80f56~tplv-xv4ileqgde-image.image","alt":"三次函数","note":[{"type":"paragraph","node_id":"llc577jza7n","attrs":{},"children":[{"type":"text","text":"三次函数"}]}],"align":"center","width":656,"height":496,"id":"7267400840804040763","version":"1","license":"","name":"三次函数","copyright":"","quote_user_id":"","should_transform":false}},{"type":"paragraph","node_id":"llc577jzkv9","attrs":{},"children":[{"type":"text","text":"其中，（"},{"type":"formula","node_id":"m0fbdy3l1tlm","attrs":{"formula_height":"3.343","formula_width":"20.495","image_id":"7346108004116414502","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/64bf94b186f24494a4c2d62748aa1d7f~tplv-xv4ileqgde-image.image","value":"A>0，－10"}},{"type":"text","text":"时，方程有一个实根和一对"},{"type":"internal_link","node_id":"m0ul6ldm21hf","attrs":{"docId":"4133960538478345113","target":1,"title":"共轭","doc_id":"4133960538478345113"},"children":[{"type":"text","text":"共轭"}]},{"type":"text","text":"虚根。"}]},{"type":"paragraph","node_id":"llc577jz1h2v","attrs":{},"children":[{"type":"text","text":"当"},{"type":"formula","node_id":"m0fbdy3l1u78","attrs":{"formula_height":"3.509","formula_width":"18.952","image_id":"7346108027432353843","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/e3e587d70ac5441c874993388a56ead9~tplv-xv4ileqgde-image.image","value":"Δ=B^2 －4AC=0"}},{"type":"text","text":"时，方程有三个实根，其中有一个二重根。"}]},{"type":"paragraph","node_id":"llc577jz295","attrs":{},"children":[{"type":"text","text":"当"},{"type":"formula","node_id":"m0fbdy3l1do8","attrs":{"formula_height":"3.509","formula_width":"18.952","image_id":"7346108032419364891","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/ac3391ca0ce74ecb9f6ffbbc9a69d885~tplv-xv4ileqgde-image.image","value":"Δ=B^2 －4AC<0"}},{"type":"text","text":"时，方程有三个不"},{"type":"internal_link","node_id":"ltwgxr6d21y2","attrs":{"docId":"4025607702353038229","target":1,"title":"相等","doc_id":"4025607702353038229"},"children":[{"type":"text","text":"相等"}]},{"type":"text","text":"的实根。"}]},{"type":"paragraph","node_id":"llc577jznev","attrs":{},"children":[{"type":"text","marks":[{"type":"strong"}],"text":"3.盛金"},{"type":"internal_link","node_id":"lmzok2mxc6x","attrs":{"docId":"8607972194790743201","target":1,"title":"定理","doc_id":"8607972194790743201"},"children":[{"type":"text","text":"定理"}],"marks":[{"type":"strong"}]}]},{"type":"paragraph","node_id":"llc577jz214i","attrs":{},"children":[{"type":"text","text":"当"},{"type":"formula","node_id":"m0fbdy3llb2","attrs":{"formula_height":"3.343","formula_width":"12.693","image_id":"7346103969653112884","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/7626967e6a894993a074b8c74d736b93~tplv-xv4ileqgde-image.image","value":"b=0，c=0"}},{"type":"text","text":"时，盛金公式1无意义；当"},{"type":"formula","node_id":"m0fbdy3l10ff","attrs":{"formula_height":"2.176","formula_width":"6.004","image_id":"7346103620666097698","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/81ca1e669a5e449bae40d8f335d54ee9~tplv-xv4ileqgde-image.image","value":"A=0"}},{"type":"text","text":"时，盛金公式3无意义；当"},{"type":"formula","node_id":"m0fbdy3l16uy","attrs":{"formula_height":"2.343","formula_width":"6.004","image_id":"7346103512092377088","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/fa40ad6f4b474c0e9866eab54d36b4b7~tplv-xv4ileqgde-image.image","value":"A≤0"}},{"type":"text","text":"时，盛金公式4无意义；当"},{"type":"formula","node_id":"m0fbdy3lpq9","attrs":{"formula_height":"3.343","formula_width":"8.064","image_id":"7346103932785016884","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/b0e182c824bf4710a66182f31af7cf57~tplv-xv4ileqgde-image.image","value":"T<－1"}},{"type":"text","text":"或"},{"type":"formula","node_id":"m0fbdy3l1tlo","attrs":{"formula_height":"2.176","formula_width":"5.897","image_id":"7346103340784467983","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/75bd6154afbc4ff091bace6df6e14826~tplv-xv4ileqgde-image.image","value":"T>1"}},{"type":"text","text":"时，盛金公式4无意义。"}]},{"type":"paragraph","node_id":"llc577jz1ggh","attrs":{},"children":[{"type":"text","text":"当"},{"type":"formula","node_id":"m0fbdy3lrpm","attrs":{"formula_height":"3.343","formula_width":"12.693","image_id":"7346103969653112884","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/7626967e6a894993a074b8c74d736b93~tplv-xv4ileqgde-image.image","value":"b=0，c=0"}},{"type":"text","text":"时，盛金公式1是否成立？盛金公式3与盛金公式4是否存在"},{"type":"formula","node_id":"m0fbdy3l1ogb","attrs":{"formula_height":"2.343","formula_width":"6.004","image_id":"7346103512092377088","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/fa40ad6f4b474c0e9866eab54d36b4b7~tplv-xv4ileqgde-image.image","value":"A≤0"}},{"type":"text","text":"的值？盛金公式4是否存在"},{"type":"formula","node_id":"m0fbdy3l32i","attrs":{"formula_height":"3.343","formula_width":"8.064","image_id":"7346103932785016884","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/b0e182c824bf4710a66182f31af7cf57~tplv-xv4ileqgde-image.image","value":"T<－1"}},{"type":"text","text":"或"},{"type":"formula","node_id":"m0fbdy3l10ub","attrs":{"formula_height":"2.176","formula_width":"5.897","image_id":"7346103340784467983","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/75bd6154afbc4ff091bace6df6e14826~tplv-xv4ileqgde-image.image","value":"T>1"}},{"type":"text","text":"的值？盛金定理给出如下回答："}]},{"type":"paragraph","node_id":"llc577jzluz","attrs":{},"children":[{"type":"text","text":"盛金定理1：当"},{"type":"formula","node_id":"m0fbdy3l1omk","attrs":{"formula_height":"2.176","formula_width":"10.866","image_id":"7346104016075636788","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/45771b831c494b9690283082cbf0d01d~tplv-xv4ileqgde-image.image","value":"A=B=0"}},{"type":"text","text":"时，若"},{"type":"formula","node_id":"m0fbdy3l9r0","attrs":{"formula_height":"2.176","formula_width":"5.258","image_id":"7346098076156887081","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/1f5bbb7719204cf48d68b54470bad993~tplv-xv4ileqgde-image.image","value":"b=0"}},{"type":"text","text":"，则必定有"},{"type":"formula","node_id":"m0fbdy3l27v","attrs":{"formula_height":"2.176","formula_width":"9.582","image_id":"7346104016075653172","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/0b2224940f5a448aa33fae384722e78e~tplv-xv4ileqgde-image.image","value":"c=d=0"}},{"type":"text","text":"（此时，"},{"type":"internal_link","node_id":"ltwgxr6d1v2f","attrs":{"docId":"2293567924838015583","target":1,"title":"方程","doc_id":"2293567924838015583"},"children":[{"type":"text","text":"方程"}]},{"type":"text","text":"有一个三重实根0，盛金公式1仍成立）。"}]},{"type":"paragraph","node_id":"llc577jz2lg","attrs":{},"children":[{"type":"text","text":"盛金"},{"type":"internal_link","node_id":"lmzok2nw1eje","attrs":{"docId":"8607972194790743201","target":1,"title":"定理","doc_id":"8607972194790743201"},"children":[{"type":"text","text":"定理"}]},{"type":"text","text":"2：当"},{"type":"formula","node_id":"m0fbdy3lu5r","attrs":{"formula_height":"2.176","formula_width":"10.866","image_id":"7346104016075636788","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/45771b831c494b9690283082cbf0d01d~tplv-xv4ileqgde-image.image","value":"A=B=0"}},{"type":"text","text":"时，若"},{"type":"formula","node_id":"m0fbdy3l1c40","attrs":{"formula_height":"2.676","formula_width":"5.258","image_id":"7346098127771795495","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/db75de3c16d142668c5349b2cb83bc38~tplv-xv4ileqgde-image.image","value":"b≠0"}},{"type":"text","text":"，则必定有"},{"type":"formula","node_id":"m0fbdy3larc","attrs":{"formula_height":"2.676","formula_width":"5.268","image_id":"7346103873519632419","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/ff5f9633054146a7a721fe3efc997d6d~tplv-xv4ileqgde-image.image","value":"c≠0"}},{"type":"text","text":"（此时，适用盛金公式1解题）。"}]},{"type":"block_image","node_id":"m4i4g0gnw3","attrs":{"url":"//p6-sdbk2-sign.byteimg.com/tos-cn-i-ry4let6mk6/d43a3216969943af8c4b7b7089e577ff~tplv-ry4let6mk6-image.image?lk3s=56f7592b&x-expires=1733900996&x-signature=qHcYVYwGOY4Lghr6zPoLHIeRsR4%3D","alt":"盛金公式1","note":[{"type":"paragraph","node_id":"llc577jz1g2x","attrs":{},"children":[{"type":"text","text":"盛金公式1"}]}],"align":"center","width":7944,"height":3952,"id":"7446665673670967332","version":"1","license":"","name":"盛金公式1.jpeg","copyright":"no_copyright","quote_user_id":"","should_transform":false}},{"type":"paragraph","node_id":"llc577jz1odk","attrs":{},"children":[{"type":"text","text":"盛金定理3：当"},{"type":"formula","node_id":"m0fbdy3l1yll","attrs":{"formula_height":"2.176","formula_width":"10.866","image_id":"7346104016075636788","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/45771b831c494b9690283082cbf0d01d~tplv-xv4ileqgde-image.image","value":"A=B=0"}},{"type":"text","text":"时，则必定有"},{"type":"formula","node_id":"m0fbdy3lg06","attrs":{"formula_height":"2.176","formula_width":"6.027","image_id":"7346104110854193187","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/a46ea662455d4de9942cc164601ac5f6~tplv-xv4ileqgde-image.image","value":"C=0"}},{"type":"text","text":"（此时，适用盛金公式1解题）。"}]},{"type":"paragraph","node_id":"llc577jzpy5","attrs":{},"children":[{"type":"text","text":"盛金定理4：当"},{"type":"formula","node_id":"m0fbdy3l986","attrs":{"formula_height":"2.176","formula_width":"6.004","image_id":"7346103620666097698","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/81ca1e669a5e449bae40d8f335d54ee9~tplv-xv4ileqgde-image.image","value":"A=0"}},{"type":"text","text":"时，若"},{"type":"formula","node_id":"m0fbdy3l6ae","attrs":{"formula_height":"2.676","formula_width":"6.025","image_id":"7346104016075669556","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/be86b7f4508041e5a7470e220b72b8b0~tplv-xv4ileqgde-image.image","value":"B≠0"}},{"type":"text","text":"，则必定有"},{"type":"formula","node_id":"m0fbdy3l20zo","attrs":{"formula_height":"2.176","formula_width":"6.197","image_id":"7346103932776824832","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/8f96199ec8274687a5d67f7fdf98cee0~tplv-xv4ileqgde-image.image","value":"Δ>0"}},{"type":"text","text":"（此时，适用盛金公式2解题）。"}]},{"type":"paragraph","node_id":"llc577jz13qd","attrs":{},"children":[{"type":"text","text":"盛金定理5：当"},{"type":"formula","node_id":"m0fbdy3l1ve3","attrs":{"formula_height":"2.176","formula_width":"6.004","image_id":"7346104076700205107","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/ca319861a0e845aca369b905a7058de5~tplv-xv4ileqgde-image.image","value":"A<0"}},{"type":"text","text":"时，则必定有"},{"type":"formula","node_id":"m0fbdy3l16rx","attrs":{"formula_height":"2.176","formula_width":"6.197","image_id":"7346103932776824832","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/8f96199ec8274687a5d67f7fdf98cee0~tplv-xv4ileqgde-image.image","value":"Δ>0"}},{"type":"text","text":"（此时，适用盛金公式2解题）。"}]},{"type":"paragraph","node_id":"llc577jz1s61","attrs":{},"children":[{"type":"text","text":"盛金定理6：当"},{"type":"formula","node_id":"m0fbdy3l23j5","attrs":{"formula_height":"2.176","formula_width":"6.197","image_id":"7346103969653178420","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/bd35236faec24f65860f324bc1b4630c~tplv-xv4ileqgde-image.image","value":"Δ=0"}},{"type":"text","text":"时，若"},{"type":"formula","node_id":"m0fbdy3lws4","attrs":{"formula_height":"2.176","formula_width":"6.004","image_id":"7346103620666097698","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/81ca1e669a5e449bae40d8f335d54ee9~tplv-xv4ileqgde-image.image","value":"A=0"}},{"type":"text","text":"，则必定有"},{"type":"formula","node_id":"m0fbdy3lfbn","attrs":{"formula_height":"2.176","formula_width":"6.025","image_id":"7346096098353725478","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/d4f69f7cc17347a59850cf5652cf7ad6~tplv-xv4ileqgde-image.image","value":"B=0"}},{"type":"text","text":"（此时，适用盛金公式1解题）。"}]},{"type":"paragraph","node_id":"llc577jz1scg","attrs":{},"children":[{"type":"text","text":"盛金定理7：当"},{"type":"formula","node_id":"m0fbdy3ls6f","attrs":{"formula_height":"2.176","formula_width":"6.197","image_id":"7346103969653178420","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/bd35236faec24f65860f324bc1b4630c~tplv-xv4ileqgde-image.image","value":"Δ=0"}},{"type":"text","text":"时，若"},{"type":"formula","node_id":"m0fbdy3lnu","attrs":{"formula_height":"2.676","formula_width":"6.025","image_id":"7346104016075669556","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/be86b7f4508041e5a7470e220b72b8b0~tplv-xv4ileqgde-image.image","value":"B≠0"}},{"type":"text","text":"，盛金公式3一定不存在"},{"type":"formula","node_id":"m0fbdy3l1etl","attrs":{"formula_height":"2.343","formula_width":"6.004","image_id":"7346103512092377088","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/fa40ad6f4b474c0e9866eab54d36b4b7~tplv-xv4ileqgde-image.image","value":"A≤0"}},{"type":"text","text":"的值（此时，适用盛金公式3解题）。"}]},{"type":"paragraph","node_id":"llc577jz1tj5","attrs":{},"children":[{"type":"text","text":"盛金"},{"type":"internal_link","node_id":"lujupi1kkc0","attrs":{"docId":"8607972194790743201","target":1,"title":"定理","doc_id":"8607972194790743201"},"children":[{"type":"text","text":"定理"}]},{"type":"text","text":"8：当"},{"type":"formula","node_id":"m0fbdy3l226x","attrs":{"formula_height":"2.176","formula_width":"6.197","image_id":"7346103572280655924","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/168ba331dfa847c4ae51bbbafa224f9c~tplv-xv4ileqgde-image.image","value":"Δ<0"}},{"type":"text","text":"时，盛金公式4一定不存在"},{"type":"formula","node_id":"m0fbdy3l1hbw","attrs":{"formula_height":"2.343","formula_width":"6.004","image_id":"7346103512092377088","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/fa40ad6f4b474c0e9866eab54d36b4b7~tplv-xv4ileqgde-image.image","value":"A≤0"}},{"type":"text","text":"的值。（此时，适用盛金公式4解题）。"}]},{"type":"paragraph","node_id":"llc577jz1006","attrs":{},"children":[{"type":"text","text":"盛金定理9：当"},{"type":"formula","node_id":"m0fbdy3lbp4","attrs":{"formula_height":"2.176","formula_width":"6.197","image_id":"7346103572280655924","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/168ba331dfa847c4ae51bbbafa224f9c~tplv-xv4ileqgde-image.image","value":"Δ<0"}},{"type":"text","text":"时，盛金公式4一定不存在"},{"type":"formula","node_id":"m0fbdy3l1gi1","attrs":{"formula_height":"2.343","formula_width":"7.705","image_id":"7346104119939088436","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/0d062bb3954d4d73926d16bf1cee98f9~tplv-xv4ileqgde-image.image","value":"T≤-1"}},{"type":"text","text":"或"},{"type":"formula","node_id":"m0fbdy3l15vl","attrs":{"formula_height":"2.343","formula_width":"5.897","image_id":"7346103873519697955","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/8ef6f5645a754f00a1e93f1bfecc983a~tplv-xv4ileqgde-image.image","value":"T≥1"}},{"type":"text","text":"的值，即T出现的值必定是"},{"type":"formula","node_id":"m0fbdy3lga5","attrs":{"formula_height":"2.343","formula_width":"11.966","image_id":"7346104016075735092","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/cbf2132c909c48609d3b8bbf79348b34~tplv-xv4ileqgde-image.image","value":"-10"}},{"type":"text","text":"时，不一定有"},{"type":"formula","node_id":"m0fbdy3lesn","attrs":{"formula_height":"2.176","formula_width":"6.004","image_id":"7346104076700205107","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/ca319861a0e845aca369b905a7058de5~tplv-xv4ileqgde-image.image","value":"A<0"}},{"type":"text","text":"。"}]},{"type":"paragraph","node_id":"llc577jzihw","attrs":{},"children":[{"type":"text","text":"盛金定理表明：盛金公式始终保持有意义。任意实系数的"},{"type":"text","text":"一元三次方程"},{"type":"text","text":"都可以运用盛金公式直观求解。"}]},{"type":"heading","node_id":"32","attrs":{"level":2},"children":[{"type":"text","text":"卡尔丹公式法"}]},{"type":"paragraph","node_id":"llc577jz1fvn","attrs":{},"children":[{"type":"text","text":"此外，一元三次方程的"},{"type":"text","text":"求根公式"},{"type":"text","text":"用通常的演绎思维是作不出来的，用类似解"},{"type":"text","text":"一元二次方程"},{"type":"text","text":"的求根公式的"},{"type":"text","text":"配方法"},{"type":"text","text":"只能将型如"},{"type":"formula","node_id":"m0fbdy3l1zwx","attrs":{"formula_height":"2.843","formula_width":"23.33","image_id":"7346108026102841371","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/74aba3decc6c47279165855783d88389~tplv-xv4ileqgde-image.image","value":"ax ^3+bx^2 +cx+d=0"}},{"type":"text","text":"的标准型一元三次方程形式化为"},{"type":"formula","node_id":"m0fbdy3l7vt","attrs":{"formula_height":"3.009","formula_width":"15.894","image_id":"7346108012408520713","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/38a6f8ecf1894ded868b2bcdab13cd04~tplv-xv4ileqgde-image.image","value":"x^3 +px+q=0"}},{"type":"text","text":"的特殊型。"}]},{"type":"paragraph","node_id":"llc577jzpsa","attrs":{},"children":[{"type":"text","text":"上世纪80年代，中国的一名中学数学教师"},{"type":"text","text":"范盛金"},{"type":"text","text":"对解一元"},{"type":"text","text":"三次方程"},{"type":"text","text":"问题进行了深入的研究和探索，发明了比卡尔丹公式更实用的新求根公式——盛金公式，并建立了简明的、直观的、实用的新判别法——盛金判别法，同时提出了盛金"},{"type":"internal_link","node_id":"lujupi1k1yvg","attrs":{"docId":"8607972194790743201","target":1,"title":"定理","doc_id":"8607972194790743201"},"children":[{"type":"text","text":"定理"}]},{"type":"text","text":"，盛金定理清晰地回答了解三次方程的疑惑问题，且很有趣味。"}]},{"type":"paragraph","node_id":"llc577jz3p3","attrs":{},"children":[{"type":"text","text":"盛金公式的特点是由"},{"type":"text","marks":[{"type":"strong"}],"text":"最简重根判别式"},{"type":"formula","node_id":"m0fbdy3l1pwv","attrs":{"formula_height":"3.509","formula_width":"41.903","image_id":"7346108032419397659","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/aff75d152e5a49b2a8421b7d351dbfea~tplv-xv4ileqgde-image.image","value":"A=b^2－3ac；B=bc－9ad；C=c^2－3bd"}},{"type":"text","marks":[{"type":"strong"}],"text":"和总根的判别式"},{"type":"formula","node_id":"m0fbdy3l981","attrs":{"formula_height":"3.509","formula_width":"14.691","image_id":"7346102492784230435","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/564dc9b3ed7346c1830842b836f5c16a~tplv-xv4ileqgde-image.image","value":"Δ=B^2－4AC"}},{"type":"text","marks":[{"type":"strong"}],"text":"来构成"},{"type":"text","text":"，体现了数学的有序、对称、和谐与简洁美，简明易记、解题直观、准确高效，当"},{"type":"formula","node_id":"m0fbdy3l10yi","attrs":{"formula_height":"3.509","formula_width":"18.952","image_id":"7346108027432353843","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/e3e587d70ac5441c874993388a56ead9~tplv-xv4ileqgde-image.image","value":"Δ=B^2 －4AC=0"}},{"type":"text","text":"时，盛金公式3："},{"type":"formula","node_id":"m0fbdy3l16kn","attrs":{"formula_height":"5.343","formula_width":"17.189","image_id":"7346108026186760201","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/8e9b61e0d1da43118b64e413e9768126~tplv-xv4ileqgde-image.image","value":"X(1)=－\\frac{b}{a}+K"}},{"type":"text","text":"；"},{"type":"formula","node_id":"m0fbdy3lrff","attrs":{"formula_height":"5.176","formula_width":"17.712","image_id":"7346108012408537097","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/5e756ffc2d224a9597655744e9e5750f~tplv-xv4ileqgde-image.image","value":"X(2)X(3)=-\\frac{K}{2}"}},{"type":"text","text":"，其中"},{"type":"formula","node_id":"m0fbdy3ln4o","attrs":{"formula_height":"5.343","formula_width":"17.745","image_id":"7346108009732620338","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/af03fbd86ea0422dab64fc64a43c1b48~tplv-xv4ileqgde-image.image","value":"K=\\frac{B}{A}，(A≠0)"}},{"type":"text","text":"，其表达式漂亮，不存在开方（此时的卡尔丹公式仍存在开立方），手算解题效率高。盛金公式3被称为超级简便的公式。盛金公式与判别法及"},{"type":"internal_link","node_id":"ltwgxr6d17no","attrs":{"docId":"8607972194790743201","target":1,"title":"定理","doc_id":"8607972194790743201"},"children":[{"type":"text","text":"定理"}]},{"type":"text","text":"形成了一套完整的、简明的、实用的、具有数学美的解三次方程的理论体系，"},{"type":"internal_link","node_id":"ltwgxr6d117w","attrs":{"docId":"7979614391597994348","target":1,"title":"范盛金","doc_id":"7979614391597994348"},"children":[{"type":"text","text":"范盛金"}]},{"type":"text","text":"创造出的这套万能的系统方法，对研究解高次方程问题及提高解三次方程的效率作出了贡献。"}]},{"type":"paragraph","node_id":"llc577jzieq","attrs":{},"children":[{"type":"text","text":"一元三次"},{"type":"text","text":"方程"},{"type":"text","text":"的求解公式的解法只能用归纳思维得到，即根据"},{"type":"internal_link","node_id":"ltwgxr6d3s","attrs":{"docId":"7241892137804218400","target":1,"title":"一元一次方程","doc_id":"7241892137804218400"},"children":[{"type":"text","text":"一元一次方程"}]},{"type":"text","text":"、"},{"type":"internal_link","node_id":"ltwgxr6d1atm","attrs":{"docId":"7240061571770531879","target":1,"title":"一元二次方程","doc_id":"7240061571770531879"},"children":[{"type":"text","text":"一元二次方程"}]},{"type":"text","text":"及特殊的高次方程的"},{"type":"text","text":"求根公式"},{"type":"text","text":"的形式归纳出一元三次方程的求根公式的形式。归纳出来的形如"},{"type":"formula","node_id":"m0fbdy3l1x65","attrs":{"formula_height":"3.009","formula_width":"15.894","image_id":"7346108045098745894","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/4c8486d55f5945718d1c68f07fb734b9~tplv-xv4ileqgde-image.image","value":" x^3 +px+q=0"}},{"type":"text","text":"的"},{"type":"text","text":"一元三次方程"},{"type":"text","text":"的求根公式的形式应该为"},{"type":"formula","node_id":"m0fbdy3lj2g","attrs":{"formula_height":"3.676","formula_width":"16.806","image_id":"7346108026102906907","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/74aa69a1052e414bae5bb38bf3478df0~tplv-xv4ileqgde-image.image","value":"x=A^{(\\frac{1}{3})}+B^{(\\frac{1}{3})}"}},{"type":"text","text":"型，即为两个开立方之和。归纳出了"},{"type":"internal_link","node_id":"m0ul6ldm22h5","attrs":{"docId":"7559204565483686457","target":1,"title":"一元三次方程求根公式","doc_id":"7559204565483686457"},"children":[{"type":"text","text":"一元三次方程求根公式"}]},{"type":"text","text":"的形式，下一步的工作就是求出开立方里的内容，也就是用p和q表示A和B。"}]},{"type":"paragraph","node_id":"llc577jzm6v","attrs":{},"children":[{"type":"text","text":"方法如下："}]},{"type":"paragraph","node_id":"llc577jz16wu","attrs":{},"children":[{"type":"text","text":"⑴将"},{"type":"formula","node_id":"m0fbdy3l182s","attrs":{"formula_height":"3.676","formula_width":"16.806","image_id":"7346108010256973851","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/2c47deb7b55c431fb387be1556b2e8b3~tplv-xv4ileqgde-image.image","value":"x=A^{(\\frac{1}{3})} +B ^{(\\frac{1}{3})}"}},{"type":"text","text":"两边同时立方可以得到"}]},{"type":"paragraph","node_id":"llc577jz22gf","attrs":{},"children":[{"type":"text","text":"⑵"},{"type":"formula","node_id":"m0fbdy3l1zbl","attrs":{"formula_height":"4.009","formula_width":"40.161","image_id":"7346108046000554035","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/0a64e03c7ed341e89a22aef9b143cebb~tplv-xv4ileqgde-image.image","value":"x^3 =(A+B)+3(AB)^{(\\frac{1}{3})} (A^{(\\frac{1}{3})} +B^{(\\frac{1}{3})} )"}}]},{"type":"paragraph","node_id":"llc577jzojh","attrs":{},"children":[{"type":"text","text":"⑶由于"},{"type":"formula","node_id":"m0fbdy3l1kgd","attrs":{"formula_height":"3.676","formula_width":"16.806","image_id":"7346108055043457074","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/d729c6dfcdeb4153aa2f50fff8b92af4~tplv-xv4ileqgde-image.image","value":"x=A^{(\\frac{1}{3})} +B^{(\\frac{1}{3})}"}},{"type":"text","text":" ，所以⑵可化为 "},{"type":"formula","node_id":"m0fbdy3lzo7","attrs":{"formula_height":"4.009","formula_width":"27.303","image_id":"7346108060319875099","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/60eb696b07a743d387cdc6510034ec9a~tplv-xv4ileqgde-image.image","value":"x^3 =(A+B)+3(AB)^{(\\frac{1}{3})} x"}},{"type":"text","text":"，移项可得"}]},{"type":"paragraph","node_id":"llc577jz5j4","attrs":{},"children":[{"type":"text","text":"⑷"},{"type":"formula","node_id":"m0fbdy3lc4j","attrs":{"formula_height":"4.343","formula_width":"31.221","image_id":"7346108046000619571","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/f79b413f1ba64c1bb01d8271698097da~tplv-xv4ileqgde-image.image","value":"x^3 －3(AB)^{(\\frac{1}{3})} x－（A+B)=0"}},{"type":"text","text":"，和"},{"type":"text","text":"一元三次方程"},{"type":"text","text":"和特殊型"},{"type":"formula","node_id":"m0fbdy3lt40","attrs":{"formula_height":"3.009","formula_width":"15.894","image_id":"7346108012408520713","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/38a6f8ecf1894ded868b2bcdab13cd04~tplv-xv4ileqgde-image.image","value":"x^3 +px+q=0"}},{"type":"text","text":"作比较，可知"}]},{"type":"paragraph","node_id":"llc577jz13to","attrs":{},"children":[{"type":"text","text":"⑸"},{"type":"formula","node_id":"m0fbdy3ldvh","attrs":{"formula_height":"4.343","formula_width":"33.849","image_id":"7346108055043506226","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/16af7df7736f456dbd3ae4c549c05b76~tplv-xv4ileqgde-image.image","value":"－3(AB)^{(\\frac{1}{3})} =p，－（A+B)=q"}},{"type":"text","text":"，化简得"}]},{"type":"paragraph","node_id":"llc577jz1pzo","attrs":{},"children":[{"type":"text","text":"⑹"},{"type":"formula","node_id":"m0fbdy3l8xv","attrs":{"formula_height":"4.843","formula_width":"28.132","image_id":"7346108045098893350","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/a2b34d37473e4181bd86288c818f42b8~tplv-xv4ileqgde-image.image","value":"A+B=－q，AB=-(\\frac{p}{3})^3"}}]},{"type":"paragraph","node_id":"llc577jz11v5","attrs":{},"children":[{"type":"text","text":"⑺这样其实就将一元三次方程的"},{"type":"text","text":"求根公式"},{"type":"text","text":"化为了"},{"type":"internal_link","node_id":"ltwgxr6d81t","attrs":{"docId":"7240061571770531879","target":1,"title":"一元二次方程","doc_id":"7240061571770531879"},"children":[{"type":"text","text":"一元二次方程"}]},{"type":"text","text":"的求根公式问题，因为A和B可以看作是一元二次方程的两个根，而⑹则是关于形如"},{"type":"formula","node_id":"m0fbdy3l1c72","attrs":{"formula_height":"3.009","formula_width":"16.546","image_id":"7346108027432648755","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/5bd8541004144e0f8411c630f5f45ebc~tplv-xv4ileqgde-image.image","value":"ay ^2+by+c=0"}},{"type":"text","text":"的一元二次方程两个根的"},{"type":"internal_link","node_id":"ltwgxr6d104c","attrs":{"docId":"7225163403723554868","target":1,"title":"韦达定理","doc_id":"7225163403723554868"},"children":[{"type":"text","text":"韦达定理"}]},{"type":"text","text":"，即"}]},{"type":"paragraph","node_id":"llc577jz1jif","attrs":{},"children":[{"type":"text","text":"⑻"},{"type":"formula","node_id":"m0fbdy3l1ycu","attrs":{"formula_height":"5.343","formula_width":"27.956","image_id":"7346108026103037979","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/52fdd4896af54eb09758911a0225f148~tplv-xv4ileqgde-image.image","value":"y_1+y_2=－\\frac{b}{a}，y_1\\cdot y_2=\\frac{c}{a}"}},{"type":"text","text":"（韦达定理）"}]},{"type":"paragraph","node_id":"llc577jzjx0","attrs":{},"children":[{"type":"text","text":"⑼对比⑹和⑻，可令"},{"type":"formula","node_id":"m0fbdy3li1u","attrs":{"formula_height":"5.343","formula_width":"39.699","image_id":"7346108074689560613","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/148490630eb94cf2943b833a026224f9~tplv-xv4ileqgde-image.image","value":"A=y_1，B=y_2，q=\\frac{b}{a}，-(\\frac{p}{3})^3=\\frac{c}{a}"}}]},{"type":"paragraph","node_id":"llc577jz6tp","attrs":{},"children":[{"type":"text","text":"⑽由于形为"},{"type":"formula","node_id":"m0fbdy3lh1m","attrs":{"formula_height":"3.009","formula_width":"16.546","image_id":"7346108060319989787","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/bc4d41f72f334d489a9bad01b63f893a~tplv-xv4ileqgde-image.image","value":"ay^2 +by+c=0"}},{"type":"text","text":"的"},{"type":"internal_link","node_id":"ltwgxr6d21dq","attrs":{"docId":"7240061571770531879","target":1,"title":"一元二次方程","doc_id":"7240061571770531879"},"children":[{"type":"text","text":"一元二次方程"}]},{"type":"text","text":"求根公式"},{"type":"text","text":"为"}]},{"type":"paragraph","node_id":"llc577jzwbe","attrs":{},"children":[{"type":"formula","node_id":"m0fbdy3l1qmj","attrs":{"formula_height":"7.676","formula_width":"21.715","image_id":"7346108046000750643","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/2efa48e0824848aaab06dafa0096bbc3~tplv-xv4ileqgde-image.image","value":"y_1=\\frac{-b+\\sqrt{b^2－4ac} }{2a}"}}]},{"type":"paragraph","node_id":"llc577jz1u4m","attrs":{},"children":[{"type":"formula","node_id":"m0fbdy3l3eu","attrs":{"formula_height":"7.676","formula_width":"21.715","image_id":"7346108058222854170","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/776efb0d23dd47f289f33549f075be73~tplv-xv4ileqgde-image.image","value":"y_2=\\frac{-b-\\sqrt{b^2－4ac} }{2a}"}}]},{"type":"paragraph","node_id":"llc577jz13ov","attrs":{},"children":[{"type":"text","text":"可化为⑾"},{"type":"formula","node_id":"m0fbdy3lxlp","attrs":{"formula_height":"6.343","formula_width":"30.468","image_id":"7346108045098975270","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/83f3a8baef664a7cb1e23e0876a23d48~tplv-xv4ileqgde-image.image","value":"y_1=－(\\frac{b}{2a})-(\\frac{b}{2a })^2 -\\sqrt{(\\frac{c}{a})}"}}]},{"type":"paragraph","node_id":"llc577jzbv7","attrs":{},"children":[{"type":"formula","node_id":"m0fbdy3lkmx","attrs":{"formula_height":"6.343","formula_width":"30.468","image_id":"7346108069274828837","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/8795543a5bcd4a6da658c53f55f0faa1~tplv-xv4ileqgde-image.image","value":"y_2=－(\\frac{b}{2a})+(\\frac{b}{2a })^2 -\\sqrt{(\\frac{c}{a})}"}}]},{"type":"paragraph","node_id":"llc577jzady","attrs":{},"children":[{"type":"text","text":"将⑼中的"},{"type":"formula","node_id":"m0fbdy3l1tfo","attrs":{"formula_height":"5.343","formula_width":"39.699","image_id":"7346108074689560613","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/148490630eb94cf2943b833a026224f9~tplv-xv4ileqgde-image.image","value":"A=y_1，B=y_2，q=\\frac{b}{a}，-(\\frac{p}{3})^3=\\frac{c}{a}"}},{"type":"text","text":"代入⑾可得"}]},{"type":"paragraph","node_id":"llc577jzi4u","attrs":{},"children":[{"type":"text","text":"⑿"},{"type":"formula","node_id":"m0fbdy3l1pdk","attrs":{"formula_height":"6.176","formula_width":"28.552","image_id":"7346108074689626149","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/193f90bc0bac499b979d280b04d67b80~tplv-xv4ileqgde-image.image","value":"A=－(\\frac{q}{2})-(\\frac{q}{2})^2 +\\sqrt{(\\frac{p}{3})^3} "}}]},{"type":"paragraph","node_id":"llc577jz1zjs","attrs":{},"children":[{"type":"formula","node_id":"m0fbdy3l1fob","attrs":{"formula_height":"6.176","formula_width":"28.573","image_id":"7346108046000783411","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/608afb20f9184018a6aa28cd6a667d8b~tplv-xv4ileqgde-image.image","value":"B=－(\\frac{q}{2})+(\\frac{q}{2})^2 +\\sqrt{(\\frac{p}{3})^3} "}}]},{"type":"paragraph","node_id":"llc577jztw8","attrs":{},"children":[{"type":"text","text":"⒀将A，B代入"},{"type":"formula","node_id":"m0fbdy3ltt1","attrs":{"formula_height":"3.676","formula_width":"14.247","image_id":"7346108074689642533","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/fbb09aab2a584eef8fbab408aff853e9~tplv-xv4ileqgde-image.image","value":"x=A ^{\\frac{1}{3}}+B ^{\\frac{1}{3}}"}},{"type":"text","text":"得"}]},{"type":"paragraph","node_id":"llc577jz13x4","attrs":{},"children":[{"type":"text","text":"⒁"},{"type":"formula","node_id":"m0fbdy3l23yy","attrs":{"formula_height":"6.176","formula_width":"58.308","image_id":"7346108046000832563","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/3123ca110a2d49bba4bb68c155f51600~tplv-xv4ileqgde-image.image","value":"x=(－(\\frac{q}{2})-(\\frac{q}{2})^2 +\\sqrt{(\\frac{p}{3})^3} )+(－(\\frac{q}{2})+(\\frac{q}{2})^2 +\\sqrt{(\\frac{p}{3})^3} )"}}]},{"type":"paragraph","node_id":"llc577jzij4","attrs":{},"children":[{"type":"text","text":"式 ⒁只是一元三方程的一个实根解，按"},{"type":"text","text":"韦达定理"},{"type":"text","text":"一元三次方程"},{"type":"text","text":"应该有三个根，不过按韦达定理一元三次方程只要求出了其中一个根，另两个根就容易求出了。"}]},{"type":"heading","node_id":"4","attrs":{"level":1},"children":[{"type":"text","text":"其他性质"}]},{"type":"paragraph","node_id":"llc577jz1u5s","attrs":{},"children":[{"type":"text","text":"特殊性质；在高考中的应用；三次函数的三大性质。"}]},{"type":"paragraph","node_id":"llc577jzfj2","attrs":{},"children":[{"type":"text","text":"详见参考资料。"}]}]