因式分解有几种解法，分别为 因式分解：把一个多项式化为几个整式的积的形式，这
\u56e0\u5f0f\u5206\u89e31\uff0e\u8fd0\u7528\u516c\u5f0f\u6cd5
\u5728\u6574\u5f0f\u7684\u4e58\u3001\u9664\u4e2d\uff0c\u6211\u4eec\u5b66\u8fc7\u82e5\u5e72\u4e2a\u4e58\u6cd5\u516c\u5f0f\uff0c\u73b0\u5c06\u5176\u53cd\u5411\u4f7f\u7528\uff0c\u5373\u4e3a\u56e0\u5f0f\u5206\u89e3\u4e2d\u5e38\u7528\u7684\u516c\u5f0f\uff0c\u4f8b\u5982\uff1a
(1)a2-b2=(a+b)(a-b)\uff1b
(2)a2\u00b12ab+b2=(a\u00b1b)2\uff1b
(3)a3+b3=(a+b)(a2-ab+b2)\uff1b
(4)a3-b3=(a-b)(a2+ab+b2)\uff0e
\u4e0b\u9762\u518d\u8865\u5145\u51e0\u4e2a\u5e38\u7528\u7684\u516c\u5f0f\uff1a
(5)a2+b2+c2+2ab+2bc+2ca=(a+b+c)2\uff1b
(6)a3+b3+c3-3abc=(a+b+c)(a2+b2+c2-ab-bc-ca)\uff1b
(7)an-bn=(a-b)(an-1+an-2b+an-3b2+\u2026+abn-2+bn-1)\u5176\u4e2dn\u4e3a\u6b63\u6574\u6570\uff1b
(8)an-bn=(a+b)(an-1-an-2b+an-3b2-\u2026+abn-2-bn-1)\uff0c\u5176\u4e2dn\u4e3a\u5076\u6570\uff1b
(9)an+bn=(a+b)(an-1-an-2b+an-3b2-\u2026-abn-2+bn-1)\uff0c\u5176\u4e2dn\u4e3a\u5947\u6570\uff0e
\u8fd0\u7528\u516c\u5f0f\u6cd5\u5206\u89e3\u56e0\u5f0f\u65f6\uff0c\u8981\u6839\u636e\u591a\u9879\u5f0f\u7684\u7279\u70b9\uff0c\u6839\u636e\u5b57\u6bcd\u3001\u7cfb\u6570\u3001\u6307\u6570\u3001\u7b26\u53f7\u7b49\u6b63\u786e\u6070\u5f53\u5730\u9009\u62e9\u516c\u5f0f\uff0e
\u4f8b1 \u5206\u89e3\u56e0\u5f0f\uff1a
(1)-2x5n-1yn+4x3n-1yn+2-2xn-1yn+4\uff1b
(2)x3-8y3-z3-6xyz\uff1b
(3)a2+b2+c2-2bc+2ca-2ab\uff1b
(4)a7-a5b2+a2b5-b7\uff0e
\u89e3 (1)\u539f\u5f0f=-2xn-1yn(x4n-2x2ny2+y4)
=-2xn-1yn[(x2n)2-2x2ny2+(y2)2]
=-2xn-1yn(x2n-y2)2
=-2xn-1yn(xn-y)2(xn+y)2\uff0e
(2)\u539f\u5f0f=x3+(-2y)3+(-z)3-3x(-2y)(-Z)
=(x-2y-z)(x2+4y2+z2+2xy+xz-2yz)\uff0e
(3)\u539f\u5f0f=(a2-2ab+b2)+(-2bc+2ca)+c2
\uff1d(a-b)2+2c(a-b)+c2
=(a-b+c)2\uff0e
\u672c\u5c0f\u9898\u53ef\u4ee5\u7a0d\u52a0\u53d8\u5f62\uff0c\u76f4\u63a5\u4f7f\u7528\u516c\u5f0f(5)\uff0c\u89e3\u6cd5\u5982\u4e0b\uff1a
\u539f\u5f0f=a2+(-b)2+c2+2(-b)c+2ca+2a(-b)
=(a-b+c)2
(4)\u539f\u5f0f=(a7-a5b2)+(a2b5-b7)
=a5(a2-b2)+b5(a2-b2)
=(a2-b2)(a5+b5)
=(a+b)(a-b)(a+b)(a4-a3b+a2b2-ab3+b4)
=(a+b)2(a-b)(a4-a3b+a2b2-ab3+b4)
\u4f8b2 \u5206\u89e3\u56e0\u5f0f\uff1aa3+b3+c3-3abc\uff0e
\u672c\u9898\u5b9e\u9645\u4e0a\u5c31\u662f\u7528\u56e0\u5f0f\u5206\u89e3\u7684\u65b9\u6cd5\u8bc1\u660e\u524d\u9762\u7ed9\u51fa\u7684\u516c\u5f0f(6)\uff0e
\u5206\u6790 \u6211\u4eec\u5df2\u7ecf\u77e5\u9053\u516c\u5f0f
(a+b)3=a3+3a2b+3ab2+b3
\u7684\u6b63\u786e\u6027\uff0c\u73b0\u5c06\u6b64\u516c\u5f0f\u53d8\u5f62\u4e3a
a3+b3=(a+b)3-3ab(a+b)\uff0e
\u8fd9\u4e2a\u5f0f\u4e5f\u662f\u4e00\u4e2a\u5e38\u7528\u7684\u516c\u5f0f\uff0c\u672c\u9898\u5c31\u501f\u52a9\u4e8e\u5b83\u6765\u63a8\u5bfc\uff0e
\u89e3 \u539f\u5f0f=(a+b)3-3ab(a+b)+c3-3abc
=\u3014(a+b)3+c3\u3015-3ab(a+b+c)
=(a+b+c)\u3014(a+b)2-c(a+b)+c2]-3ab(a+b+c)
=(a+b+c)(a2+b2+c2-ab-bc-ca)\uff0e
\u8bf4\u660e \u516c\u5f0f(6)\u662f\u4e00\u4e2a\u5e94\u7528\u6781\u5e7f\u7684\u516c\u5f0f\uff0c\u7528\u5b83\u53ef\u4ee5\u63a8\u51fa\u5f88\u591a\u6709\u7528\u7684\u7ed3\u8bba\uff0c\u4f8b\u5982\uff1a\u6211\u4eec\u5c06\u516c\u5f0f(6)\u53d8\u5f62\u4e3a
a3+b3+c3-3abc
\u663e\u7136\uff0c\u5f53a+b+c=0\u65f6\uff0c\u5219a3+b3+c3=3abc\uff1b\u5f53a+b+c\uff1e0\u65f6\uff0c\u5219a3+b3+c3-3abc\u22650\uff0c\u5373a3+b3+c3\u22653abc\uff0c\u800c\u4e14\uff0c\u5f53\u4e14\u4ec5\u5f53a=b=c\u65f6\uff0c\u7b49\u53f7\u6210\u7acb\uff0e
\u5982\u679c\u4ee4x=a3\u22650\uff0cy=b3\u22650\uff0cz=c3\u22650\uff0c\u5219\u6709
\u7b49\u53f7\u6210\u7acb\u7684\u5145\u8981\u6761\u4ef6\u662fx=y=z\uff0e\u8fd9\u4e5f\u662f\u4e00\u4e2a\u5e38\u7528\u7684\u7ed3\u8bba\uff0e
\u4f8b3 \u5206\u89e3\u56e0\u5f0f\uff1ax15+x14+x13+\u2026+x2+x+1\uff0e
\u5206\u6790 \u8fd9\u4e2a\u591a\u9879\u5f0f\u7684\u7279\u70b9\u662f\uff1a\u670916\u9879\uff0c\u4ece\u6700\u9ad8\u6b21\u9879x15\u5f00\u59cb\uff0cx\u7684\u6b21\u6570\u987a\u6b21\u9012\u51cf\u81f30\uff0c\u7531\u6b64\u60f3\u5230\u5e94\u7528\u516c\u5f0fan-bn\u6765\u5206\u89e3\uff0e
\u89e3 \u56e0\u4e3a
x16-1=(x-1)(x15+x14+x13+\u2026x2+x+1)\uff0c
\u6240\u4ee5
\u8bf4\u660e \u5728\u672c\u9898\u7684\u5206\u89e3\u8fc7\u7a0b\u4e2d\uff0c\u7528\u5230\u5148\u4e58\u4ee5(x-1)\uff0c\u518d\u9664\u4ee5(x-1)\u7684\u6280\u5de7\uff0c\u8fd9\u4e00\u6280\u5de7\u5728\u7b49\u5f0f\u53d8\u5f62\u4e2d\u5f88\u5e38\u7528\uff0e
2\uff0e\u62c6\u9879\u3001\u6dfb\u9879\u6cd5
\u56e0\u5f0f\u5206\u89e3\u662f\u591a\u9879\u5f0f\u4e58\u6cd5\u7684\u9006\u8fd0\u7b97\uff0e\u5728\u591a\u9879\u5f0f\u4e58\u6cd5\u8fd0\u7b97\u65f6\uff0c\u6574\u7406\u3001\u5316\u7b80\u5e38\u5c06\u51e0\u4e2a\u540c\u7c7b\u9879\u5408\u5e76\u4e3a\u4e00\u9879\uff0c\u6216\u5c06\u4e24\u4e2a\u4ec5\u7b26\u53f7\u76f8\u53cd\u7684\u540c\u7c7b\u9879\u76f8\u4e92\u62b5\u6d88\u4e3a\u96f6\uff0e\u5728\u5bf9\u67d0\u4e9b\u591a\u9879\u5f0f\u5206\u89e3\u56e0\u5f0f\u65f6\uff0c\u9700\u8981\u6062\u590d\u90a3\u4e9b\u88ab\u5408\u5e76\u6216\u76f8\u4e92\u62b5\u6d88\u7684\u9879\uff0c\u5373\u628a\u591a\u9879\u5f0f\u4e2d\u7684\u67d0\u4e00\u9879\u62c6\u6210\u4e24\u9879\u6216\u591a\u9879\uff0c\u6216\u8005\u5728\u591a\u9879\u5f0f\u4e2d\u6dfb\u4e0a\u4e24\u4e2a\u4ec5\u7b26\u5408\u76f8\u53cd\u7684\u9879\uff0c\u524d\u8005\u79f0\u4e3a\u62c6\u9879\uff0c\u540e\u8005\u79f0\u4e3a\u6dfb\u9879\uff0e\u62c6\u9879\u3001\u6dfb\u9879\u7684\u76ee\u7684\u662f\u4f7f\u591a\u9879\u5f0f\u80fd\u7528\u5206\u7ec4\u5206\u89e3\u6cd5\u8fdb\u884c\u56e0\u5f0f\u5206\u89e3\uff0e
\u4f8b4 \u5206\u89e3\u56e0\u5f0f\uff1ax3-9x+8\uff0e
\u5206\u6790 \u672c\u9898\u89e3\u6cd5\u5f88\u591a\uff0c\u8fd9\u91cc\u53ea\u4ecb\u7ecd\u8fd0\u7528\u62c6\u9879\u3001\u6dfb\u9879\u6cd5\u5206\u89e3\u7684\u51e0\u79cd\u89e3\u6cd5\uff0c\u6ce8\u610f\u4e00\u4e0b\u62c6\u9879\u3001\u6dfb\u9879\u7684\u76ee\u7684\u4e0e\u6280\u5de7\uff0e
\u89e3\u6cd51 \u5c06\u5e38\u6570\u98798\u62c6\u6210-1+9\uff0e
\u539f\u5f0f=x3-9x-1+9
=(x3-1)-9x+9
=(x-1)(x2+x+1)-9(x-1)
=(x-1)(x2+x-8)\uff0e
\u89e3\u6cd52 \u5c06\u4e00\u6b21\u9879-9x\u62c6\u6210-x-8x\uff0e
\u539f\u5f0f=x3-x-8x+8
=(x3-x)+(-8x+8)
=x(x+1)(x-1)-8(x-1)
=(x-1)(x2+x-8)\uff0e
\u89e3\u6cd53 \u5c06\u4e09\u6b21\u9879x3\u62c6\u62109x3-8x3\uff0e
\u539f\u5f0f=9x3-8x3-9x+8
=(9x3-9x)+(-8x3+8)
=9x(x+1)(x-1)-8(x-1)(x2+x+1)
=(x-1)(x2+x-8)\uff0e
\u89e3\u6cd54 \u6dfb\u52a0\u4e24\u9879-x2+x2\uff0e
\u539f\u5f0f=x3-9x+8
=x3-x2+x2-9x+8
=x2(x-1)+(x-8)(x-1)
=(x-1)(x2+x-8)\uff0e
\u8bf4\u660e \u7531\u6b64\u9898\u53ef\u4ee5\u770b\u51fa\uff0c\u7528\u62c6\u9879\u3001\u6dfb\u9879\u7684\u65b9\u6cd5\u5206\u89e3\u56e0\u5f0f\u65f6\uff0c\u8981\u62c6\u54ea\u4e9b\u9879\uff0c\u6dfb\u4ec0\u4e48\u9879\u5e76\u65e0\u4e00\u5b9a\u4e4b\u89c4\uff0c\u4e3b\u8981\u7684\u662f\u8981\u4f9d\u9760\u5bf9\u9898\u76ee\u7279\u70b9\u7684\u89c2\u5bdf\uff0c\u7075\u6d3b\u53d8\u6362\uff0c\u56e0\u6b64\u62c6\u9879\u3001\u6dfb\u9879\u6cd5\u662f\u56e0\u5f0f\u5206\u89e3\u8bf8\u65b9\u6cd5\u4e2d\u6280\u5de7\u6027\u6700\u5f3a\u7684\u4e00\u79cd\uff0e
\u4f8b5 \u5206\u89e3\u56e0\u5f0f\uff1a
(1)x9+x6+x3-3\uff1b
(2)(m2-1)(n2-1)+4mn\uff1b
(3)(x+1)4+(x2-1)2+(x-1)4\uff1b
(4)a3b-ab3+a2+b2+1\uff0e
\u89e3 (1)\u5c06-3\u62c6\u6210-1-1-1\uff0e
\u539f\u5f0f=x9+x6+x3-1-1-1
=(x9-1)+(x6-1)+(x3-1)
=(x3-1)(x6+x3+1)+(x3-1)(x3+1)+(x3-1)
=(x3-1)(x6+2x3+3)
=(x-1)(x2+x+1)(x6+2x3+3)\uff0e
(2)\u5c064mn\u62c6\u62102mn+2mn\uff0e
\u539f\u5f0f=(m2-1)(n2-1)+2mn+2mn
=m2n2-m2-n2+1+2mn+2mn
=(m2n2+2mn+1)-(m2-2mn+n2)
=(mn+1)2-(m-n)2
=(mn+m-n+1)(mn-m+n+1)\uff0e
(3)\u5c06(x2-1)2\u62c6\u62102(x2-1)2-(x2-1)2\uff0e
\u539f\u5f0f=(x+1)4+2(x2-1)2-(x2-1)2+(x-1)4
=\u3014(x+1)4+2(x+1)2(x-1)2+(x-1)4]-(x2-1)2
=\u3014(x+1)2+(x-1)2]2-(x2-1)2
=(2x2+2)2-(x2-1)2=(3x2+1)(x2+3)\uff0e
(4)\u6dfb\u52a0\u4e24\u9879+ab-ab\uff0e
\u539f\u5f0f=a3b-ab3+a2+b2+1+ab-ab
=(a3b-ab3)+(a2-ab)+(ab+b2+1)
=ab(a+b)(a-b)+a(a-b)+(ab+b2+1)
=a(a-b)\u3014b(a+b)+1]+(ab+b2+1)
=[a(a-b)+1](ab+b2+1)
=(a2-ab+1)(b2+ab+1)\uff0e
\u8bf4\u660e (4)\u662f\u4e00\u9053\u8f83\u96be\u7684\u9898\u76ee\uff0c\u7531\u4e8e\u5206\u89e3\u540e\u7684\u56e0\u5f0f\u7ed3\u6784\u8f83\u590d\u6742\uff0c\u6240\u4ee5\u4e0d\u6613\u60f3\u5230\u6dfb\u52a0+ab-ab\uff0c\u800c\u4e14\u6dfb\u52a0\u9879\u540e\u5206\u6210\u7684\u4e09\u9879\u7ec4\u53c8\u65e0\u516c\u56e0\u5f0f\uff0c\u800c\u662f\u5148\u5c06\u524d\u4e24\u7ec4\u5206\u89e3\uff0c\u518d\u4e0e\u7b2c\u4e09\u7ec4\u7ed3\u5408\uff0c\u627e\u5230\u516c\u56e0\u5f0f\uff0e\u8fd9\u9053\u9898\u76ee\u4f7f\u6211\u4eec\u4f53\u4f1a\u5230\u62c6\u9879\u3001\u6dfb\u9879\u6cd5\u7684\u6781\u5f3a\u6280\u5de7\u6240\u5728\uff0c\u540c\u5b66\u4eec\u9700\u591a\u505a\u7ec3\u4e60\uff0c\u79ef\u7d2f\u7ecf\u9a8c\uff0e
3\uff0e\u6362\u5143\u6cd5
\u6362\u5143\u6cd5\u6307\u7684\u662f\u5c06\u4e00\u4e2a\u8f83\u590d\u6742\u7684\u4ee3\u6570\u5f0f\u4e2d\u7684\u67d0\u4e00\u90e8\u5206\u770b\u4f5c\u4e00\u4e2a\u6574\u4f53\uff0c\u5e76\u7528\u4e00\u4e2a\u65b0\u7684\u5b57\u6bcd\u66ff\u4ee3\u8fd9\u4e2a\u6574\u4f53\u6765\u8fd0\u7b97\uff0c\u4ece\u800c\u4f7f\u8fd0\u7b97\u8fc7\u7a0b\u7b80\u660e\u6e05\u6670\uff0e
\u4f8b6 \u5206\u89e3\u56e0\u5f0f\uff1a(x2+x+1)(x2+x+2)-12\uff0e
\u5206\u6790 \u5c06\u539f\u5f0f\u5c55\u5f00\uff0c\u662f\u5173\u4e8ex\u7684\u56db\u6b21\u591a\u9879\u5f0f\uff0c\u5206\u89e3\u56e0\u5f0f\u8f83\u56f0\u96be\uff0e\u6211\u4eec\u4e0d\u59a8\u5c06x2+x\u770b\u4f5c\u4e00\u4e2a\u6574\u4f53\uff0c\u5e76\u7528\u5b57\u6bcdy\u6765\u66ff\u4ee3\uff0c\u4e8e\u662f\u539f\u9898\u8f6c\u5316\u4e3a\u5173\u4e8ey\u7684\u4e8c\u6b21\u4e09\u9879\u5f0f\u7684\u56e0\u5f0f\u5206\u89e3\u95ee\u9898\u4e86\uff0e
\u89e3 \u8bbex2+x=y\uff0c\u5219
\u539f\u5f0f=(y+1)(y+2)-12=y2+3y-10
=(y-2)(y+5)=(x2+x-2)(x2+x+5)
=(x-1)(x+2)(x2+x+5)\uff0e
\u8bf4\u660e \u672c\u9898\u4e5f\u53ef\u5c06x2+x+1\u770b\u4f5c\u4e00\u4e2a\u6574\u4f53\uff0c\u6bd4\u5982\u4ecax2+x+1=u\uff0c\u4e00\u6837\u53ef\u4ee5\u5f97\u5230\u540c\u6837\u7684\u7ed3\u679c\uff0c\u6709\u5174\u8da3\u7684\u540c\u5b66\u4e0d\u59a8\u8bd5\u4e00\u8bd5\uff0e
\u4f8b7 \u5206\u89e3\u56e0\u5f0f\uff1a
(x2+3x+2)(4x2+8x+3)-90\uff0e
\u5206\u6790 \u5148\u5c06\u4e24\u4e2a\u62ec\u53f7\u5185\u7684\u591a\u9879\u5f0f\u5206\u89e3\u56e0\u5f0f\uff0c\u7136\u540e\u518d\u91cd\u65b0\u7ec4\u5408\uff0e
\u89e3 \u539f\u5f0f=(x+1)(x+2)(2x+1)(2x+3)-90
=[(x+1)(2x+3)][(x+2)(2x+1)]-90
=(2x2+5x+3)(2x2+5x+2)-90\uff0e
\u4ee4y=2x2+5x+2\uff0c\u5219
\u539f\u5f0f=y(y+1)-90=y2+y-90
=(y+10)(y-9)
=(2x2+5x+12)(2x2+5x-7)
=(2x2+5x+12)(2x+7)(x-1)\uff0e
\u8bf4\u660e \u5bf9\u591a\u9879\u5f0f\u9002\u5f53\u7684\u6052\u7b49\u53d8\u5f62\u662f\u6211\u4eec\u627e\u5230\u65b0\u5143(y)\u7684\u57fa\u7840\uff0e
\u4f8b8 \u5206\u89e3\u56e0\u5f0f\uff1a
(x2+4x+8)2+3x(x2+4x+8)+2x2\uff0e
\u89e3 \u8bbex2+4x+8=y\uff0c\u5219
\u539f\u5f0f=y2+3xy+2x2=(y+2x)(y+x)
=(x2+6x+8)(x2+5x+8)
=(x+2)(x+4)(x2+5x+8)\uff0e
\u8bf4\u660e \u7531\u672c\u9898\u53ef\u77e5\uff0c\u7528\u6362\u5143\u6cd5\u5206\u89e3\u56e0\u5f0f\u65f6\uff0c\u4e0d\u5fc5\u5c06\u539f\u5f0f\u4e2d\u7684\u5143\u90fd\u7528\u65b0\u5143\u4ee3\u6362\uff0c\u6839\u636e\u9898\u76ee\u9700\u8981\uff0c\u5f15\u5165\u5fc5\u8981\u7684\u65b0\u5143\uff0c\u539f\u5f0f\u4e2d\u7684\u53d8\u5143\u548c\u65b0\u53d8\u5143\u53ef\u4ee5\u4e00\u8d77\u53d8\u5f62\uff0c\u6362\u5143\u6cd5\u7684\u672c\u8d28\u662f\u7b80\u5316\u591a\u9879\u5f0f\uff0e
\u4f8b9 \u5206\u89e3\u56e0\u5f0f\uff1a6x4+7x3-36x2-7x+6\uff0e
\u89e3\u6cd51 \u539f\u5f0f=6(x4+1)\uff0b7x(x2-1)-36x2
=6\u3014(x4-2x2+1)+2x2\u3015+7x(x2-1)-36x2
=6[(x2-1)2+2x2]+7x(x2-1)-36x2
=6(x2-1)2+7x(x2-1)-24x2
=[2(x2-1)-3x\u3015\u30143(x2-1)+8x]
=(2x2-3x-2)(3x2+8x-3)
=(2x+1)(x-2)(3x-1)(x+3)\uff0e
\u8bf4\u660e \u672c\u89e3\u6cd5\u5b9e\u9645\u4e0a\u662f\u5c06x2-1\u770b\u4f5c\u4e00\u4e2a\u6574\u4f53\uff0c\u4f46\u5e76\u6ca1\u6709\u8bbe\u7acb\u65b0\u5143\u6765\u4ee3\u66ff\u5b83\uff0c\u5373\u719f\u7ec3\u4f7f\u7528\u6362\u5143\u6cd5\u540e\uff0c\u5e76\u975e\u6bcf\u9898\u90fd\u8981\u8bbe\u7f6e\u65b0\u5143\u6765\u4ee3\u66ff\u6574\u4f53\uff0e
\u89e3\u6cd52
\u539f\u5f0f=x2[6(t2+2)+7t-36]
=x2(6t2+7t-24)=x2(2t-3)(3t+8)
=x2[2(x-1/x)-3][3(x-1/x)+8]
=(2x2-3x-2)(3x2+8x-3)
=(2x+1)(x-2)(3x-1)(x+3)\uff0e
\u4f8b10 \u5206\u89e3\u56e0\u5f0f\uff1a(x2+xy+y2)-4xy(x2+y2)\uff0e
\u5206\u6790 \u672c\u9898\u542b\u6709\u4e24\u4e2a\u5b57\u6bcd\uff0c\u4e14\u5f53\u4e92\u6362\u8fd9\u4e24\u4e2a\u5b57\u6bcd\u7684\u4f4d\u7f6e\u65f6\uff0c\u591a\u9879\u5f0f\u4fdd\u6301\u4e0d\u53d8\uff0c\u8fd9\u6837\u7684\u591a\u9879\u5f0f\u53eb\u4f5c\u4e8c\u5143\u5bf9\u79f0\u5f0f\uff0e\u5bf9\u4e8e\u8f83\u96be\u5206\u89e3\u7684\u4e8c\u5143\u5bf9\u79f0\u5f0f\uff0c\u7ecf\u5e38\u4ee4u=x+y\uff0cv=xy\uff0c\u7528\u6362\u5143\u6cd5\u5206\u89e3\u56e0\u5f0f\uff0e
\u89e3 \u539f\u5f0f=[(x+y)2-xy]2-4xy[(x+y)2-2xy]\uff0e\u4ee4x+y=u\uff0cxy=v\uff0c\u5219
\u539f\u5f0f=(u2-v)2-4v(u2-2v)
=u4-6u2v+9v2
=(u2-3v)2
=(x2+2xy+y2-3xy)2
=(x2-xy+y2)2\uff0e
1\uff0e\u53cc\u5341\u5b57\u76f8\u4e58\u6cd5
\u5206\u89e3\u4e8c\u6b21\u4e09\u9879\u5f0f\u65f6\uff0c\u6211\u4eec\u5e38\u7528\u5341\u5b57\u76f8\u4e58\u6cd5\uff0e\u5bf9\u4e8e\u67d0\u4e9b\u4e8c\u5143\u4e8c\u6b21\u516d\u9879\u5f0f(ax2+bxy+cy2+dx+ey+f)\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u7528\u5341\u5b57\u76f8\u4e58\u6cd5\u5206\u89e3\u56e0\u5f0f\uff0e
\u4f8b\u5982\uff0c\u5206\u89e3\u56e0\u5f0f2x2-7xy-22y2-5x+35y-3\uff0e\u6211\u4eec\u5c06\u4e0a\u5f0f\u6309x\u964d\u5e42\u6392\u5217\uff0c\u5e76\u628ay\u5f53\u4f5c\u5e38\u6570\uff0c\u4e8e\u662f\u4e0a\u5f0f\u53ef\u53d8\u5f62\u4e3a
2x2-(5+7y)x-(22y2-35y+3)\uff0c
\u53ef\u4ee5\u770b\u4f5c\u662f\u5173\u4e8ex\u7684\u4e8c\u6b21\u4e09\u9879\u5f0f\uff0e
\u5bf9\u4e8e\u5e38\u6570\u9879\u800c\u8a00\uff0c\u5b83\u662f\u5173\u4e8ey\u7684\u4e8c\u6b21\u4e09\u9879\u5f0f\uff0c\u4e5f\u53ef\u4ee5\u7528\u5341\u5b57\u76f8\u4e58\u6cd5\uff0c\u5206\u89e3\u4e3a
\u5373
-22y2+35y-3=(2y-3)(-11y+1)\uff0e
\u518d\u5229\u7528\u5341\u5b57\u76f8\u4e58\u6cd5\u5bf9\u5173\u4e8ex\u7684\u4e8c\u6b21\u4e09\u9879\u5f0f\u5206\u89e3
\u6240\u4ee5
\u539f\u5f0f=\u3014x+(2y-3)\u3015\u30142x+(-11y+1)\u3015
=(x+2y-3)(2x-11y+1)\uff0e
\u4e0a\u8ff0\u56e0\u5f0f\u5206\u89e3\u7684\u8fc7\u7a0b\uff0c\u5b9e\u65bd\u4e86\u4e24\u6b21\u5341\u5b57\u76f8\u4e58\u6cd5\uff0e\u5982\u679c\u628a\u8fd9\u4e24\u4e2a\u6b65\u9aa4\u4e2d\u7684\u5341\u5b57\u76f8\u4e58\u56fe\u5408\u5e76\u5728\u4e00\u8d77\uff0c\u53ef\u5f97\u5230\u4e0b\u56fe\uff1a
\u5b83\u8868\u793a\u7684\u662f\u4e0b\u9762\u4e09\u4e2a\u5173\u7cfb\u5f0f\uff1a
(x+2y)(2x-11y)=2x2-7xy-22y2\uff1b
(x-3)(2x+1)=2x2-5x-3\uff1b
(2y-3)(-11y+1)=-22y2+35y-3\uff0e
\u8fd9\u5c31\u662f\u6240\u8c13\u7684\u53cc\u5341\u5b57\u76f8\u4e58\u6cd5\uff0e
\u7528\u53cc\u5341\u5b57\u76f8\u4e58\u6cd5\u5bf9\u591a\u9879\u5f0fax2+bxy+cy2+dx+ey+f\u8fdb\u884c\u56e0\u5f0f\u5206\u89e3\u7684\u6b65\u9aa4\u662f\uff1a
(1)\u7528\u5341\u5b57\u76f8\u4e58\u6cd5\u5206\u89e3ax2+bxy+cy2\uff0c\u5f97\u5230\u4e00\u4e2a\u5341\u5b57\u76f8\u4e58\u56fe(\u6709\u4e24\u5217)\uff1b
(2)\u628a\u5e38\u6570\u9879f\u5206\u89e3\u6210\u4e24\u4e2a\u56e0\u5f0f\u586b\u5728\u7b2c\u4e09\u5217\u4e0a\uff0c\u8981\u6c42\u7b2c\u4e8c\u3001\u7b2c\u4e09\u5217\u6784\u6210\u7684\u5341\u5b57\u4ea4\u53c9\u4e4b\u79ef\u7684\u548c\u7b49\u4e8e\u539f\u5f0f\u4e2d\u7684ey\uff0c\u7b2c\u4e00\u3001\u7b2c\u4e09\u5217\u6784\u6210\u7684\u5341\u5b57\u4ea4\u53c9\u4e4b\u79ef\u7684\u548c\u7b49\u4e8e\u539f\u5f0f\u4e2d\u7684dx\uff0e
\u4f8b1 \u5206\u89e3\u56e0\u5f0f\uff1a
(1)x2-3xy-10y2+x+9y-2\uff1b
(2)x2-y2+5x+3y+4\uff1b
(3)xy+y2+x-y-2\uff1b
(4)6x2-7xy-3y2-xz+7yz-2z2\uff0e
\u89e3 (1)
\u539f\u5f0f=(x-5y+2)(x+2y-1)\uff0e
(2)
\u539f\u5f0f=(x+y+1)(x-y+4)\uff0e
(3)\u539f\u5f0f\u4e2d\u7f3ax2\u9879\uff0c\u53ef\u628a\u8fd9\u4e00\u9879\u7684\u7cfb\u6570\u770b\u62100\u6765\u5206\u89e3\uff0e
\u539f\u5f0f=(y+1)(x+y-2)\uff0e
(4)
\u539f\u5f0f=(2x-3y+z)(3x+y-2z)\uff0e
\u8bf4\u660e (4)\u4e2d\u6709\u4e09\u4e2a\u5b57\u6bcd\uff0c\u89e3\u6cd5\u4ecd\u4e0e\u524d\u9762\u7684\u7c7b\u4f3c\uff0e
2\uff0e\u6c42\u6839\u6cd5
\u6211\u4eec\u628a\u5f62\u5982anxn+an-1xn-1+\u2026+a1x+a0(n\u4e3a\u975e\u8d1f\u6574\u6570)\u7684\u4ee3\u6570\u5f0f\u79f0\u4e3a\u5173\u4e8ex\u7684\u4e00\u5143\u591a\u9879\u5f0f\uff0c\u5e76\u7528f(x)\uff0cg(x)\uff0c\u2026\u7b49\u8bb0\u53f7\u8868\u793a\uff0c\u5982
f(x)=x2-3x+2\uff0cg(x)=x5+x2+6\uff0c\u2026\uff0c
\u5f53x=a\u65f6\uff0c\u591a\u9879\u5f0ff(x)\u7684\u503c\u7528f(a)\u8868\u793a\uff0e\u5982\u5bf9\u4e0a\u9762\u7684\u591a\u9879\u5f0ff(x)
f(1)=12-3\u00d71+2=0\uff1b
f(-2)=(-2)2-3\u00d7(-2)+2=12\uff0e
\u82e5f(a)=0\uff0c\u5219\u79f0a\u4e3a\u591a\u9879\u5f0ff(x)\u7684\u4e00\u4e2a\u6839\uff0e
\u5b9a\u74061(\u56e0\u5f0f\u5b9a\u7406) \u82e5a\u662f\u4e00\u5143\u591a\u9879\u5f0ff(x)\u7684\u6839\uff0c\u5373f(a)=0\u6210\u7acb\uff0c\u5219\u591a\u9879\u5f0ff(x)\u6709\u4e00\u4e2a\u56e0\u5f0fx-a\uff0e
\u6839\u636e\u56e0\u5f0f\u5b9a\u7406\uff0c\u627e\u51fa\u4e00\u5143\u591a\u9879\u5f0ff(x)\u7684\u4e00\u6b21\u56e0\u5f0f\u7684\u5173\u952e\u662f\u6c42\u591a\u9879\u5f0ff(x)\u7684\u6839\uff0e\u5bf9\u4e8e\u4efb\u610f\u591a\u9879\u5f0ff(x)\uff0c\u8981\u6c42\u51fa\u5b83\u7684\u6839\u662f\u6ca1\u6709\u4e00\u822c\u65b9\u6cd5\u7684\uff0c\u7136\u800c\u5f53\u591a\u9879\u5f0ff(x)\u7684\u7cfb\u6570\u90fd\u662f\u6574\u6570\u65f6\uff0c\u5373\u6574\u7cfb\u6570\u591a\u9879\u5f0f\u65f6\uff0c\u7ecf\u5e38\u7528\u4e0b\u9762\u7684\u5b9a\u7406\u6765\u5224\u5b9a\u5b83\u662f\u5426\u6709\u6709\u7406\u6839\uff0e
\u5b9a\u74062
\u7684\u6839\uff0c\u5219\u5fc5\u6709p\u662fa0\u7684\u7ea6\u6570\uff0cq\u662fan\u7684\u7ea6\u6570\uff0e\u7279\u522b\u5730\uff0c\u5f53a0=1\u65f6\uff0c\u6574\u7cfb\u6570\u591a\u9879\u5f0ff(x)\u7684\u6574\u6570\u6839\u5747\u4e3aan\u7684\u7ea6\u6570\uff0e
\u6211\u4eec\u6839\u636e\u4e0a\u8ff0\u5b9a\u7406\uff0c\u7528\u6c42\u591a\u9879\u5f0f\u7684\u6839\u6765\u786e\u5b9a\u591a\u9879\u5f0f\u7684\u4e00\u6b21\u56e0\u5f0f\uff0c\u4ece\u800c\u5bf9\u591a\u9879\u5f0f\u8fdb\u884c\u56e0\u5f0f\u5206\u89e3\uff0e
\u4f8b2 \u5206\u89e3\u56e0\u5f0f\uff1ax3-4x2+6x-4\uff0e
\u5206\u6790 \u8fd9\u662f\u4e00\u4e2a\u6574\u7cfb\u6570\u4e00\u5143\u591a\u9879\u5f0f\uff0c\u539f\u5f0f\u82e5\u6709\u6574\u6570\u6839\uff0c\u5fc5\u662f-4\u7684\u7ea6\u6570\uff0c\u9010\u4e2a\u68c0\u9a8c-4\u7684\u7ea6\u6570\uff1a\u00b11\uff0c\u00b12\uff0c\u00b14\uff0c\u53ea\u6709
f(2)=23-4\u00d722+6\u00d72-4=0\uff0c
\u5373x=2\u662f\u539f\u5f0f\u7684\u4e00\u4e2a\u6839\uff0c\u6240\u4ee5\u6839\u636e\u5b9a\u74061\uff0c\u539f\u5f0f\u5fc5\u6709\u56e0\u5f0fx-2\uff0e
\u89e3\u6cd51 \u7528\u5206\u7ec4\u5206\u89e3\u6cd5\uff0c\u4f7f\u6bcf\u7ec4\u90fd\u6709\u56e0\u5f0f(x-2)\uff0e
\u539f\u5f0f=(x3-2x2)-(2x2-4x)+(2x-4)
=x2(x-2)-2x(x-2)+2(x-2)
=(x-2)(x2-2x+2)\uff0e
\u89e3\u6cd52 \u7528\u591a\u9879\u5f0f\u9664\u6cd5\uff0c\u5c06\u539f\u5f0f\u9664\u4ee5(x-2)\uff0c
\u6240\u4ee5
\u539f\u5f0f=(x-2)(x2-2x+2)\uff0e
\u8bf4\u660e \u5728\u4e0a\u8ff0\u89e3\u6cd5\u4e2d\uff0c\u7279\u522b\u8981\u6ce8\u610f\u7684\u662f\u591a\u9879\u5f0f\u7684\u6709\u7406\u6839\u4e00\u5b9a\u662f-4\u7684\u7ea6\u6570\uff0c\u53cd\u4e4b\u4e0d\u6210\u7acb\uff0c\u5373-4\u7684\u7ea6\u6570\u4e0d\u4e00\u5b9a\u662f\u591a\u9879\u5f0f\u7684\u6839\uff0e\u56e0\u6b64\uff0c\u5fc5\u987b\u5bf9-4\u7684\u7ea6\u6570\u9010\u4e2a\u4ee3\u5165\u591a\u9879\u5f0f\u8fdb\u884c\u9a8c\u8bc1\uff0e
\u4f8b3 \u5206\u89e3\u56e0\u5f0f\uff1a9x4-3x3+7x2-3x-2\uff0e
\u5206\u6790 \u56e0\u4e3a9\u7684\u7ea6\u6570\u6709\u00b11\uff0c\u00b13\uff0c\u00b19\uff1b-2\u7684\u7ea6\u6570\u6709\u00b11\uff0c
\u00b1
\u4e3a\uff1a
\u6240\u4ee5\uff0c\u539f\u5f0f\u6709\u56e0\u5f0f9x2-3x-2\uff0e
\u89e3 9x4-3x3+7x2-3x-2
=9x4-3x3-2x2+9x2-3x-2
\u4e3a=x2(9x3-3x-2)+9x2-3x-2
=(9x2-3x-2)(x2+1)
=(3x+1)(3x-2)(x2+1)
\u8bf4\u660e \u82e5\u6574\u7cfb\u6570\u591a\u9879\u5f0f\u6709\u5206\u6570\u6839\uff0c\u53ef\u5c06\u6240\u5f97\u51fa\u7684\u542b\u6709\u5206\u6570\u7684\u56e0\u5f0f\u5316\u4e3a\u6574\u7cfb\u6570\u56e0\u5f0f\uff0c\u5982\u4e0a\u9898\u4e2d\u7684\u56e0\u5f0f
\u53ef\u4ee5\u5316\u4e3a9x2-3x-2\uff0c\u8fd9\u6837\u53ef\u4ee5\u7b80\u5316\u5206\u89e3\u8fc7\u7a0b\uff0e
\u603b\u4e4b\uff0c\u5bf9\u4e00\u5143\u9ad8\u6b21\u591a\u9879\u5f0ff(x)\uff0c\u5982\u679c\u80fd\u627e\u5230\u4e00\u4e2a\u4e00\u6b21\u56e0\u5f0f(x-a)\uff0c\u90a3\u4e48f(x)\u5c31\u53ef\u4ee5\u5206\u89e3\u4e3a(x-a)g(x)\uff0c\u800cg(x)\u662f\u6bd4f(x)\u4f4e\u4e00\u6b21\u7684\u4e00\u5143\u591a\u9879\u5f0f\uff0c\u8fd9\u6837\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u7ee7\u7eed\u5bf9g(x)\u8fdb\u884c\u5206\u89e3\u4e86\uff0e
3\uff0e\u5f85\u5b9a\u7cfb\u6570\u6cd5
\u5f85\u5b9a\u7cfb\u6570\u6cd5\u662f\u6570\u5b66\u4e2d\u7684\u4e00\u79cd\u91cd\u8981\u7684\u89e3\u9898\u65b9\u6cd5\uff0c\u5e94\u7528\u5f88\u5e7f\u6cdb\uff0c\u8fd9\u91cc\u4ecb\u7ecd\u5b83\u5728\u56e0\u5f0f\u5206\u89e3\u4e2d\u7684\u5e94\u7528\uff0e
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\u4f8b4 \u5206\u89e3\u56e0\u5f0f\uff1ax2+3xy+2y2+4x+5y+3\uff0e
\u5206\u6790 \u7531\u4e8e
(x2+3xy+2y2)=(x+2y)(x+y)\uff0c
\u82e5\u539f\u5f0f\u53ef\u4ee5\u5206\u89e3\u56e0\u5f0f\uff0c\u90a3\u4e48\u5b83\u7684\u4e24\u4e2a\u4e00\u6b21\u9879\u4e00\u5b9a\u662fx+2y+m\u548cx\uff0by\uff0bn\u7684\u5f62\u5f0f\uff0c\u5e94\u7528\u5f85\u5b9a\u7cfb\u6570\u6cd5\u5373\u53ef\u6c42\u51fam\u548cn\uff0c\u4f7f\u95ee\u9898\u5f97\u5230\u89e3\u51b3\uff0e
\u89e3 \u8bbe
x2+3xy+2y2+4x+5y+3
=(x+2y+m)(x+y+n)
=x2+3xy+2y2+(m+n)x+(m+2n)y+mn\uff0c
\u6bd4\u8f83\u4e24\u8fb9\u5bf9\u5e94\u9879\u7684\u7cfb\u6570\uff0c\u5219\u6709
\u89e3\u4e4b\u5f97m=3\uff0cn=1\uff0e\u6240\u4ee5
\u539f\u5f0f=(x+2y+3)(x+y+1)\uff0e
\u8bf4\u660e \u672c\u9898\u4e5f\u53ef\u7528\u53cc\u5341\u5b57\u76f8\u4e58\u6cd5\uff0c\u8bf7\u540c\u5b66\u4eec\u81ea\u5df1\u89e3\u4e00\u4e0b\uff0e
\u4f8b5 \u5206\u89e3\u56e0\u5f0f\uff1ax4-2x3-27x2-44x+7\uff0e
\u5206\u6790 \u672c\u9898\u6240\u7ed9\u7684\u662f\u4e00\u5143\u6574\u7cfb\u6570\u591a\u9879\u5f0f\uff0c\u6839\u636e\u524d\u9762\u8bb2\u8fc7\u7684\u6c42\u6839\u6cd5\uff0c\u82e5\u539f\u5f0f\u6709\u6709\u7406\u6839\uff0c\u5219\u53ea\u53ef\u80fd\u662f\u00b11\uff0c\u00b17(7\u7684\u7ea6\u6570)\uff0c\u7ecf\u68c0\u9a8c\uff0c\u5b83\u4eec\u90fd\u4e0d\u662f\u539f\u5f0f\u7684\u6839\uff0c\u6240\u4ee5\uff0c\u5728\u6709\u7406\u6570\u96c6\u5185\uff0c\u539f\u5f0f\u6ca1\u6709\u4e00\u6b21\u56e0\u5f0f\uff0e\u5982\u679c\u539f\u5f0f\u80fd\u5206\u89e3\uff0c\u53ea\u80fd\u5206\u89e3\u4e3a(x2+ax+b)(x2+cx+d)\u7684\u5f62\u5f0f\uff0e
\u89e3 \u8bbe
\u539f\u5f0f=(x2+ax+b)(x2+cx+d)
=x4+(a+c)x3+(b+d+ac)x2+(ad+bc)x+bd\uff0c
\u6240\u4ee5\u6709
\u7531bd=7\uff0c\u5148\u8003\u8651b=1\uff0cd=7\u6709
\u6240\u4ee5
\u539f\u5f0f=(x2-7x+1)(x2+5x+7)\uff0e
\u8bf4\u660e \u7531\u4e8e\u56e0\u5f0f\u5206\u89e3\u7684\u552f\u4e00\u6027\uff0c\u6240\u4ee5\u5bf9b=-1\uff0cd=-7\u7b49\u53ef\u4ee5\u4e0d\u52a0\u4ee5\u8003\u8651\uff0e\u672c\u9898\u5982\u679cb=1\uff0cd=7\u4ee3\u5165\u65b9\u7a0b\u7ec4\u540e\uff0c\u65e0\u6cd5\u786e\u5b9aa\uff0cc\u7684\u503c\uff0c\u5c31\u5fc5\u987b\u5c06bd=7\u7684\u5176\u4ed6\u89e3\u4ee3\u5165\u65b9\u7a0b\u7ec4\uff0c\u76f4\u5230\u6c42\u51fa\u5f85\u5b9a\u7cfb\u6570\u4e3a\u6b62\uff0e
\u672c\u9898\u6ca1\u6709\u4e00\u6b21\u56e0\u5f0f\uff0c\u56e0\u800c\u65e0\u6cd5\u8fd0\u7528\u6c42\u6839\u6cd5\u5206\u89e3\u56e0\u5f0f\uff0e\u4f46\u5229\u7528\u5f85\u5b9a\u7cfb\u6570\u6cd5\uff0c\u4f7f\u6211\u4eec\u627e\u5230\u4e86\u4e8c\u6b21\u56e0\u5f0f\uff0e\u7531\u6b64\u53ef\u89c1\uff0c\u5f85\u5b9a\u7cfb\u6570\u6cd5\u5728\u56e0\u5f0f\u5206\u89e3\u4e2d\u4e5f\u6709\u7528\u6b66\u4e4b\u5730\uff0e
\u56e0\u5f0f\u5206\u89e3\uff1a\u628a\u4e00\u4e2a\u591a\u9879\u5f0f\u5316\u4e3a\u51e0\u4e2a\u6574\u5f0f\u7684\u79ef\u7684\u5f62\u5f0f\uff0c\u8fd9\u79cd\u53d8\u5f62\u53eb\u505a\u628a\u8fd9\u4e2a\u591a\u9879\u5f0f\u56e0\u5f0f\u5206\u89e3,\u4e5f\u53eb\u4f5c\u5206\u89e3\u56e0\u5f0f\u3002
因式分解有几种解法，分别为一.提取公因式法；
二.运用公式法；
三."十字相乘"法；
四.分组分解法．
因式分解有几种解法，分别为：
(1)提取公因式法；
(2)分组分解法；
(3)十字相乘法；
(4)应用公式法；
(5)综合除法；
提公因式法，公式法，十字相乘法
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