因式分解练习题及答案 因式分解练习题及答案，求！！！

\u56e0\u5f0f\u5206\u89e3\u7ec3\u4e60\u9898\u53ca\u7b54\u6848

\u4f60\u53ef\u771f\u72e0 \u8981\u8fd9\u4e48\u591a\u9898\u76ee

1. 5ax+5bx+3ay+3by
\u89e3\u6cd5\uff1a=5x(a+b)+3y(a+b)
=(5x+3y)(a+b)
2. x^3-x^2+x-1
\u89e3\u6cd5\uff1a=(x^3-x^2)+(x-1)
=x^2(x-1)+ (x-1)
=(x-1)(x^2+1)
3. x2-x-y2-y
\u89e3\u6cd5\uff1a=(x2-y2)-(x+y)
=(x+y)(x-y)-(x+y)
=(x+y)(x-y-1)

bc(b+c)+ca(c-a)-ab(a+b)
=bc(c-a+a+b)+ca(c-a)-ab(a+b)
=bc(c-a)+bc(a+b)+ca(c-a)-ab(a+b)
=bc(c-a)+ca(c-a)+bc(a+b)-ab(a+b)
=(bc+ca)(c-a)+(bc-ab)(a+b)
=c(c-a)(b+a)+b(a+b)(c-a)
=(c+b)(c-a)(a+b)\uff0e


x^2+3x-40
=x^2+3x+2.25-42.25
=(x+1.5)^2-(6.5)^2
=(x+8)(x-5)\uff0e

(x^2+x+1)(x^2+x+2)-12\u65f6\uff0c\u53ef\u4ee5\u4ee4y=x^2+x,\u5219
\u539f\u5f0f=(y+1)(y+2)-12
=y^2+3y+2-12=y^2+3y-10
=(y+5)(y-2)
=(x^2+x+5)(x^2+x-2)
=(x^2+x+5)(x+2)(x-1)\uff0e

(1+y)^2-2x^2(1+y^2)+x^4(1-y)^2

\u89e3\uff1a\u539f\u5f0f=(1+y)^2+2(1+y)x^2(1+y)+x^4(1-y)^2-2(1+y)x^2(1-y)-2x^2(1+y^2)

=[(1+y)+x^2(1-y)]^2-2(1+y)x^2(1-y)-2x^2(1+y^2)

=[(1+y)+x^2(1-y)]^2-(2x)^2

=[(1+y)+x^2(1-y)+2x]\u00b7[(1+y)+x^2(1-y)-2x]

=(x^2-x^2y+2x+y+1)(x^2-x^2y-2x+y+1)

=[(x+1)^2-y(x^2-1)][(x-1)^2-y(x^2-1)]

=(x+1)(x+1-xy+y)(x-1)(x-1-xy-y)

x^5+3x^4y-5x^3y^2+4xy^4+12y^5

\u89e3\uff1a\u539f\u5f0f=(x^5+3x^4y)-(5x^3y^2+15x^2y^3)+(4xy^4+12y^5)

=x^4(x+3y)-5x^2y^2(x+3y)+4y^4(x+3y)

=(x+3y)(x^4-5x^2y^2+4y^4)

=(x+3y)(x^2-4y^2)(x^2-y^2)

=(x+3y)(x+y)(x-y)(x+2y)(x-2y)

\u5206\u89e3\u56e0\u5f0fm +5n-mn-5m
\u89e3\uff1am +5n-mn-5m= m -5m -mn+5n
= (m -5m )+(-mn+5n)
=m(m-5)-n(m-5)
=(m-5)(m-n)

\u5206\u89e3\u56e0\u5f0fbc(b+c)+ca(c-a)-ab(a+b)
\u89e3\uff1abc(b+c)+ca(c-a)-ab(a+b)=bc(c-a+a+b)+ca(c-a)-ab(a+b)
=bc(c-a)+ca(c-a)+bc(a+b)-ab(a+b)
=c(c-a)(b+a)+b(a+b)(c-a)
=(c+b)(c-a)(a+b)

1.(2a-b)²+8ab
2.y²-2y-x²+1
3.x²-xy+yz-xz
4.6x²+5x-4
5.2a²-7ab+6b²
6.(x²-2x)²+2(x²-2x)+1
7.(x²-2x)²-14(x²-2x)-15
8.x²(x-y)+(y-x)
9.169(a+b)²-121(a-b)²
10.(x-3)(x-5)+1
\u7b54\u6848\uff1a1.(2a-b)²+8ab=(2a+b)²
2.y²-2y-x²+1=(y-1)²-x²=(y-1-x)(y-1+x)
3.x²-xy+yz-xz =x(x-y)-z(x-y)=(x-z)(x-y)
4.6x²+5x-4 =(2x-1)(3x+4)
5.2a²-7ab+6b²=(2a-3b)(a-2b)
6.(x²-2x)²+2(x²-2x)+1 =(x²-2x+1)²=(x-1)^4
7.(x²-2x)²-14(x²-2x)-15 =(x²-2x-15)(x²-2x+1)=(x+3)(x-5)(x-1)²
8.x²(x-y)+(y-x) =(x²-1)(x-y)=(x+1)(x-1)(x-y)
9.169(a+b)²-121(a-b)²
=(14a+14b-11a+11b)(14a+14b+11a-11b)
=(3a+25b)(25a+3b)
10.(x-3)(x-5)+1 =(x-3)²-2(x-3)+1 =(x-3-1)²=(x-4)²

-5a^2+16a=a(16-5a)
8x^2-4x=4x(2x-1)
15p+10p^2\uff1d5p(3+2p)
\uff0d3x^2y-6xy=-3xy(x+2y)
14m^3n^2-6m^2n^3=2m^2n^2(7m-6n)
27a^2 b^3 c+18ab^2=9ab^2(3abc+2)
18xy^2 z^3+12x^2 y^2=6xy^2(3z^3+2x)
8m^2 n^2 -6m^3 n^2=2m^2 n^2(4-3m)

\u56e0\u5f0f\u5206\u89e33a3b2c\uff0d6a2b2c2\uff0b9ab2c3\uff1d3ab^2 c(a^2-2ac+3c^2)
3.\u56e0\u5f0f\u5206\u89e3xy\uff0b6\uff0d2x\uff0d3y\uff1d(x-3)(y-2)
4.\u56e0\u5f0f\u5206\u89e3x2(x\uff0dy)\uff0by2(y\uff0dx)\uff1d(x+y)(x-y)^2
5.\u56e0\u5f0f\u5206\u89e32x2\uff0d(a\uff0d2b)x\uff0dab\uff1d(2x-a)(x+b)
6.\u56e0\u5f0f\u5206\u89e3a4\uff0d9a2b2\uff1da^2(a+3b)(a-3b)
7.\u82e5\u5df2\u77e5x3\uff0b3x2\uff0d4\u542b\u6709x\uff0d1\u7684\u56e0\u5f0f\uff0c\u8bd5\u5206\u89e3x3\uff0b3x2\uff0d4\uff1d(x-1)(x+2)^2
8.\u56e0\u5f0f\u5206\u89e3ab(x2\uff0dy2)\uff0bxy(a2\uff0db2)\uff1d(ay+bx)(ax-by)
9.\u56e0\u5f0f\u5206\u89e3(x\uff0by)(a\uff0db\uff0dc)\uff0b(x\uff0dy)(b\uff0bc\uff0da)\uff1d2y(a-b-c)
10.\u56e0\u5f0f\u5206\u89e3a2\uff0da\uff0db2\uff0db\uff1d(a+b)(a-b-1)
11.\u56e0\u5f0f\u5206\u89e3(3a\uff0db)2\uff0d4(3a\uff0db)(a\uff0b3b)\uff0b4(a\uff0b3b)2\uff1d[3a-b-2(a+3b)]^2=(a-7b)^2
12.\u56e0\u5f0f\u5206\u89e3(a\uff0b3)2\uff0d6(a\uff0b3)\uff1d(a+3)(a-3)
13.\u56e0\u5f0f\u5206\u89e3(x\uff0b1)2(x\uff0b2)\uff0d(x\uff0b1)(x\uff0b2)2\uff1d-(x+1)(x+2) abc\uff0bab\uff0d4a\uff1da(bc+b-4)
(2)16x2\uff0d81\uff1d(4x+9)(4x-9)
(3)9x2\uff0d30x\uff0b25\uff1d(3x-5)^2
(4)x2\uff0d7x\uff0d30\uff1d(x-10)(x+3)
35.\u56e0\u5f0f\u5206\u89e3x2\uff0d25\uff1d(x+5)(x-5)
36.\u56e0\u5f0f\u5206\u89e3x2\uff0d20x\uff0b100\uff1d(x-10)^2
37.\u56e0\u5f0f\u5206\u89e3x2\uff0b4x\uff0b3\uff1d(x+1)(x+3)
38.\u56e0\u5f0f\u5206\u89e34x2\uff0d12x\uff0b5\uff1d(2x-1)(2x-5)
39.\u56e0\u5f0f\u5206\u89e3\u4e0b\u5217\u5404\u5f0f\uff1a (1)3ax2\uff0d6ax\uff1d3ax(x-2) (2)x(x\uff0b2)\uff0dx\uff1dx(x+1) (3)x2\uff0d4x\uff0dax\uff0b4a\uff1d(x-4)(x-a) (4)25x2\uff0d49\uff1d(5x-9)(5x+9) (5)36x2\uff0d60x\uff0b25\uff1d(6x-5)^2 (6)4x2\uff0b12x\uff0b9\uff1d(2x+3)^2 (7)x2\uff0d9x\uff0b18\uff1d(x-3)(x-6) (8)2x2\uff0d5x\uff0d3\uff1d(x-3)(2x+1) (9)12x2\uff0d50x\uff0b8\uff1d2(6x-1)(x-4)
40.\u56e0\u5f0f\u5206\u89e3(x\uff0b2)(x\uff0d3)\uff0b(x\uff0b2)(x\uff0b4)\uff1d(x+2)(2x-1)
41.\u56e0\u5f0f\u5206\u89e32ax2\uff0d3x\uff0b2ax\uff0d3\uff1d (x+1)(2ax-3)
42.\u56e0\u5f0f\u5206\u89e39x2\uff0d66x\uff0b121\uff1d(3x-11)^2
43.\u56e0\u5f0f\u5206\u89e38\uff0d2x2\uff1d2(2+x)(2-x)
44.\u56e0\u5f0f\u5206\u89e3x2\uff0dx\uff0b14 \uff1d\u6574\u6570\u5185\u65e0\u6cd5\u5206\u89e3
45.\u56e0\u5f0f\u5206\u89e39x2\uff0d30x\uff0b25\uff1d(3x-5)^2
46.\u56e0\u5f0f\u5206\u89e3\uff0d20x2\uff0b9x\uff0b20\uff1d(-4x+5)(5x+4)
47.\u56e0\u5f0f\u5206\u89e312x2\uff0d29x\uff0b15\uff1d(4x-3)(3x-5)
48.\u56e0\u5f0f\u5206\u89e336x2\uff0b39x\uff0b9\uff1d3(3x+1)(4x+3)
49.\u56e0\u5f0f\u5206\u89e321x2\uff0d31x\uff0d22\uff1d(21x+11)(x-2)
50.\u56e0\u5f0f\u5206\u89e39x4\uff0d35x2\uff0d4\uff1d(9x^2+1)(x+2)(x-2)
51.\u56e0\u5f0f\u5206\u89e3(2x\uff0b1)(x\uff0b1)\uff0b(2x\uff0b1)(x\uff0d3)\uff1d2(x-1)(2x+1)
52.\u56e0\u5f0f\u5206\u89e32ax2\uff0d3x\uff0b2ax\uff0d3\uff1d(x+1)(2ax-3)
53.\u56e0\u5f0f\u5206\u89e3x(y\uff0b2)\uff0dx\uff0dy\uff0d1\uff1d(x-1)(y+1)
54.\u56e0\u5f0f\u5206\u89e3(x2\uff0d3x)\uff0b(x\uff0d3)2\uff1d(x-3)(2x-3)
55.\u56e0\u5f0f\u5206\u89e39x2\uff0d66x\uff0b121\uff1d(3x-11)^2
56.\u56e0\u5f0f\u5206\u89e38\uff0d2x2\uff1d2(2-x)(2+x)
57.\u56e0\u5f0f\u5206\u89e3x4\uff0d1\uff1d(x-1)(x+1)(x^2+1)
58.\u56e0\u5f0f\u5206\u89e3x2\uff0b4x\uff0dxy\uff0d2y\uff0b4\uff1d(x+2)(x-y+2)
59.\u56e0\u5f0f\u5206\u89e34x2\uff0d12x\uff0b5\uff1d(2x-1)(2x-5)
60.\u56e0\u5f0f\u5206\u89e321x2\uff0d31x\uff0d22\uff1d(21x+11)(x-2)
61.\u56e0\u5f0f\u5206\u89e34x2\uff0b4xy\uff0by2\uff0d4x\uff0d2y\uff0d3\uff1d(2x+y-3)(2x+y+1)
62.\u56e0\u5f0f\u5206\u89e39x5\uff0d35x3\uff0d4x\uff1dx(9x^2+1)(x+2)(x-2)

140\uff0em2(p-q)-p+q
141\uff0e(2m+3n)(2m-n)-4n(2m-n)\uff0e
142\uff0e(x+2y)2-x2-2xy\uff0e
143\uff0ea(ab+bc+ac)-abc\uff0e
144\uff0eab-a-b+1\uff0e
145\uff0exyz-xy-xz+x-yz+y+z-1\uff0e
146\uff0ex4-2y4-2x3y+xy3\uff0e
148\uff0eabc(a2+b2+c2)-a3bc+2ab2c2\uff0e
149\uff0e(a-b-c)(a+b-c)-(b-c-a)(b+c-a)\uff0e
150\uff0ea2(b-c)+b2(c-a)+c2(a-b)\uff0e
a(a+b)(a-b)-a(a+b)2\uff0e
152\uff0e(x2-2x)2+2x(x-2)+1\uff0e
153\uff0e2acd-c2a-ad2\uff0e
154\uff0e(x-y)2+12(y-x)z+36z2\uff0e
156\uff0ex2-4ax+8ab-4b2\uff0e
157\uff0ea2b2+c2d2-2abcd+2ab-2cd+1\uff0e
158\uff0e(ax+by)2+(ay-bx)2+2(ax+by)(ay-bx)\uff0e
160\uff0e(1-a2)(1-b2)-(a2-1)2(b2-1)2\uff0e
161\uff0e3x4-48y4\uff0e
162\uff0e(x+1)2-9(x-1)2\uff0e
163\uff0e(x2+pq)2-(p+q)2x2\uff0e
164\uff0e(1+2xy)2-(x2+y2)2\uff0e
165\uff0e4a2b2-(a2+b2)2\uff0e
166\uff0e4a2b2-(a2+b2-c2)2\uff0e
167\uff0e(c2-a2-b2)2-4a2b2\uff0e
168\uff0e(x2-b2+y2-a2)2-4(ab-xy)2\uff0e
169\uff0e64a4(x+1)2-49b4(x+1)4\uff0e
170\uff0eab2-ac2+4ac-4a\uff0e
171\uff0e4a2-c2+6ab+3bc\uff0e
172\uff0ex3n+y3n\uff0e
174\uff0e(x+y)3+125\uff0e
176\uff0e(m-n)6-(m+n)6\uff0e
177\uff0e(x+1)6-(x-1)6\uff0e
178\uff0ea12-b12\uff0e
179\uff0e(z-x)3-(y+z)3\uff0e
180\uff0e(3m-2n)3+(3m+2n)3\uff0e
181\uff0ex6(x2-y2)+y6(y2-x2)\uff0e
182\uff0e8(x+y)3+1\uff0e
183\uff0e(a-1)3-(b+1)3\uff0e
184\uff0e(a+b+c)3-a3-b3-c3\uff0e
185\uff0ex2+4xy+3y2\uff0e
186\uff0ex2y2-18xy+65\uff0e
187\uff0ex2+18x-144\uff0e
188\uff0ex2+30x+144\uff0e
189\uff0ex4+2x2-8\uff0e
190\uff0e3x4+6x2-9\uff0e
191\uff0e-m4+18m2-17\uff0e
192\uff0e3x4-7x2y2-20y4\uff0e
193\uff0ex5-2x3-8x\uff0e
194\uff0ea3-5a2b-300ab2\uff0e
195\uff0ex8+19x5-216x2\uff0e
196\uff0e6a4n+k-a2n+k-35ak\uff0e
199\uff0e30x2+8xy-182y2\uff0e
200\uff0em4+14m2-15\uff0e
140\uff0e\uff08p-q\uff09\uff08m-1\uff09\uff08m+1\uff09\uff0e
141\uff0e\uff082m-n\uff092\uff0e
142\uff0e2y\uff08x+2y\uff09\uff0e
143\uff0ea2\uff08b+c\uff09\uff0e
144\uff0e\uff08b-1\uff09\uff08a-1\uff09\uff0e
145\uff0e\uff08x-1\uff09\uff08y-1\uff09\uff08z-1\uff09\uff0e
\u63d0\u793a\uff1a\u65b9\u6cd5\u4e00 \u539f\u5f0f=x\uff08yz-y-z+1\uff09-\uff08yz-y-z+1\uff09
=\uff08yz-y-z+1\uff09\uff08x-1\uff09
=[y\uff08z-1\uff09-\uff08z-1\uff09]\uff08x-1\uff09
=\uff08x-1\uff09\uff08y-1\uff09\uff08z-1\uff09\uff0e
\u65b9\u6cd5\u4e8c \u539f\u5f0f=xy\uff08z-1\uff09-x\uff08z-1\uff09-y\uff08z-1\uff09+\uff08z-1\uff09
=\uff08xy-x-y+1\uff09\uff08z-1\uff09
=[x\uff08y-1\uff09-\uff08y-1\uff09]\uff08z-1\uff09
=\uff08x-1\uff09\uff08y-1\uff09\uff08z-1\uff09\uff0e
146\uff0e\uff08x-2y\uff09\uff08x+y\uff09\uff08x2-xy+y2\uff09\uff0e
\u63d0\u793a\uff1a\u65b9\u6cd5\u4e00 \u539f\u5f0f=x\uff08x3+y3\uff09-2y\uff08x3+y3\uff09
=\uff08x3+y3\uff09\uff08x-2y\uff09
=\uff08x-2y\uff09\uff08x+y\uff09\uff08x2-xy+y2\uff09\uff0e
\u65b9\u6cd5\u4e8c \u539f\u5f0f=x3\uff08x-2y\uff09+y3\uff08x-2y\uff09
=\uff08x-2y\uff09\uff08x3+y3\uff09
=\uff08x-2y\uff09\uff08x+y\uff09\uff08x2-xy+y2\uff09\uff0e
148\uff0eabc\uff08b+c\uff092\uff0e
\u63d0\u793a\uff1a\u539f\u5f0f=abc\uff08a2+b2+c2-a2+2bc\uff09\uff0e
149\uff0e2\uff08b-c\uff09\uff08a-b-c\uff09\uff0e
\u63d0\u793a\uff1a\u539f\u5f0f=\uff08a-b-c\uff09\uff08a+b-c\uff09+\uff08a-b-c\uff09\uff08b-c-a\uff09
=\uff08a-b-c\uff09[\uff08a+b-c\uff09+\uff08b-c-a\uff09]
=2\uff08b-c\uff09\uff08a-b-c\uff09\uff0e
150\uff0e\uff08a-b\uff09\uff08b-c\uff09\uff08a-c\uff09\uff0e
\u63d0\u793a\uff1a\u539f\u5f0f=a2b-a2c+b2c-ab2+c2\uff08a-b\uff09
=\uff08a2b-ab2\uff09-\uff08a2c-b2c\uff09+c2\uff08a-b\uff09
=ab\uff08a-b\uff09-c\uff08a2-b2\uff09+c2\uff08a-b\uff09
=\uff08a-b\uff09[ab-c\uff08a+b\uff09+c2]
=\uff08a-b\uff09[a\uff08b-c\uff09-c\uff08b-c\uff09]
=\uff08a-b\uff09\uff08b-c\uff09\uff08a-c\uff09\uff0e
\u63d0\u793a\uff1a\u539f\u5f0f=a\uff08a+b\uff09[a-b-\uff08a+b\uff09]=a\uff08a+b\uff09\uff08-2b\uff09
=-2ab\uff08a+b\uff09\uff1b
152\uff0e\uff08x-1\uff094\uff0e
\u63d0\u793a\uff1a\u539f\u5f0f=[x\uff08x-2\uff09]2+2•x\uff08x-2\uff09+12
=[x\uff08x-2\uff09+1]2=\uff08x2-2x+1\uff092
=\uff08x-1\uff094\uff0e
153\uff0e-a\uff08c-d\uff092\uff0e
154\uff0e\uff08x-y-6z\uff092\uff0e
156\uff0e\uff08x-2b\uff09\uff08x-4a+2b\uff09\uff0e
157\uff0e\uff08ab-cd+1\uff092\uff0e
\u63d0\u793a\uff1a\u539f\u5f0f=\uff08a2b2-2abcd+c2d2\uff09+2\uff08ab-cd\uff09+1
=\uff08ab-cd\uff092+2\uff08ab-cd\uff09+1
=\uff08ab-cd+1\uff092\uff0e
158\uff0e\uff08ax+by+ay-bx\uff092\uff0e
160\uff0e\uff081+a\uff09\uff081-a\uff09\uff081+b\uff09\uff081-b\uff09\uff08a2+b2-a2b2\uff09\uff0e
161\uff0e3\uff08x2+4y2\uff09\uff08x+2y\uff09\uff08x-2y\uff09\uff0e
162\uff0e4\uff082x-1\uff09\uff082-x\uff09\uff0e
163\uff0e\uff08x2+px+qx+pq\uff09\uff08x2-px-qx+pq\uff09\uff0e
164\uff0e\uff081+x-y\uff09\uff081-x+y\uff09\uff08x2+y2+2xy+1\uff09\uff0e
165\uff0e-\uff08a+b\uff092\uff08a-b\uff092\uff0e
166\uff0e\uff08a+b+c\uff09\uff08a+b-c\uff09\uff08c+a-b\uff09\uff08c-a+b\uff09\uff0e
\u63d0\u793a\uff1a\u539f\u5f0f=\uff082ab+a2+b2-c2\uff09\uff082ab-a2-b2+c2\uff09
=[\uff08a+b\uff092-c2][c2-\uff08a-b\uff092]
=\uff08a+b+c\uff09\uff08a+b-c\uff09\uff08c+a-b\uff09\uff08c-a+b\uff09\uff0e
167\uff0e\uff08c+a-b\uff09\uff08c-a+b\uff09\uff08c+a+b\uff09\uff08c-a-b\uff09\uff0e
168\uff0e\uff08x+y+a+b\uff09\uff08x+y-a-b\uff09\uff08x-y+a-b\uff09\uff08x-y-a+b\uff09\uff0e
\u63d0\u793a\uff1a\u539f\u5f0f=\uff08x2-b2+y2-a2+2ab-2xy\uff09\uff08x2-b2+y2-a2-2ab+2xy\uff09
=[\uff08x2-2xy+y2\uff09-\uff08a2-2ab+b2\uff09][\uff08x2+2xy+y2\uff09
-\uff08a2+2ab+b2\uff09]
=[\uff08x-y\uff092-\uff08a-b\uff092][\uff08x+y\uff092-\uff08a+b\uff092]
=\uff08x-y+a-b\uff09\uff08x-y-a+b\uff09\uff08x+y+a+b\uff09\uff08x+y-a-b\uff09\uff0e
169\uff0e\uff08x+1\uff092\uff088a2+7b2x+7b2\uff09\uff088a2-7b2x-7b2\uff09\uff0e
170\uff0ea\uff08b-c+2\uff09\uff08b+c-2\uff09\uff0e
\u63d0\u793a\uff1a\u539f\u5f0f=a\uff08b2-c2+4c-4\uff09
=a\uff08b2-c2+2b-2b+2c+2c-4\uff09
=a[\uff08b-c\uff09\uff08b+c\uff09-2\uff08b-c\uff09+2\uff08b+c\uff09-4]
=a[\uff08b-c\uff09+2][\uff08b+c\uff09-2]\uff0e
171\uff0e\uff082a+c\uff09\uff082a-c+3b\uff09\uff0e
172\uff0e\uff08xn+yn\uff09\uff08x2n-xnyn+y2n\uff09\uff0e
174\uff0e\uff08x+y+5\uff09\uff08x2+2xy+y2-5x-5y+25\uff09\uff0e
176\uff0e-4mn\uff083n2+m2\uff09\uff083m2+n2\uff09\uff0e
\u63d0\u793a\uff1a\u539f\u5f0f=[\uff08m-n\uff093]2-[\uff08m+n\uff093]2
=[\uff08m-n\uff093+\uff08m+n\uff093][\uff08m-n\uff093-\uff08m+n\uff093]
=2m[\uff08m-n\uff092-\uff08m-n\uff09\uff08m+n\uff09\uff08m+n\uff092]
\u00d7{-2n[\uff08m-n\uff092+\uff08m-n\uff09\uff08m+n\uff09+\uff08m+n\uff092]}
=-4mn\uff08m2+3n2\uff09\uff083m2+n2\uff09\uff0e
177\uff0e4x\uff08x2+3\uff09\uff083x2+1\uff09\uff0e
\u63d0\u793a\uff1a\u539f\u5f0f=[\uff08x+1\uff093]2-[\uff08x-1\uff093]2
=[\uff08x+1\uff093+\uff08x-1\uff093][\uff08x+1\uff093-\uff08x-1\uff093]
=2x[\uff08x+1\uff092-\uff08x+1\uff09\uff08x-1\uff09+\uff08x-1\uff092]
\u00d72[\uff08x+1\uff092+\uff08x+1\uff09\uff08x-1\uff09+\uff08x-1\uff092]
=4x\uff08x2+3\uff09\uff083x2+1\uff09\uff0e
178\uff0e\uff08a-b\uff09\uff08a+b\uff09\uff08a2+b2\uff09\uff08a2+ab+b2\uff09\uff08a2-ab+b2\uff09\uff08a4-a2b2+b4\uff09\uff0e
\u63d0\u793a\uff1a\u539f\u5f0f=\uff08a6\uff092-\uff08b6\uff092=\uff08a6+b6\uff09\uff08a6-b6\uff09
=[\uff08a2\uff093+\uff08b2\uff093][\uff08a3\uff092-\uff08b3\uff092]
=\uff08a2+b2\uff09\uff08a4-a2b2+b4\uff09\uff08a3+b3\uff09\uff08a3-b3\uff09\uff0e
179\uff0e-\uff08x+y\uff09\uff08x2+y2+3z2-xy+3yz-3xz\uff09\uff0e
180\uff0e18m\uff083m2+4n2\uff09\uff0e
181\uff0e\uff08x+y\uff092\uff08x-y\uff092\uff08x2-xy+y2\uff09\uff08x2+xy+y2\uff09\uff0e
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=\uff08x+y\uff09\uff08x-y\uff09\uff08x3+y3\uff09\uff08x3-y3\uff09\uff0e
182\uff0e\uff082x+2y+1\uff09\uff084x2+8xy+4y2-2x-2y+1\uff09\uff0e
183\uff0e\uff08a-b-2\uff09\uff08a2+ab+b2-a+b+1\uff09\uff0e
184\uff0e3\uff08b+c\uff09\uff08a+b\uff09\uff08c+a\uff09\uff0e
\u63d0\u793a\uff1a\u539f\u5f0f=[\uff08a+b+c\uff093-a3]-\uff08b3+c3\uff09\uff0e
185\uff0e\uff08x+3y\uff09\uff08x+y\uff09\uff0e
186\uff0e\uff08xy-13\uff09\uff08xy-5\uff09\uff0e
187\uff0e\uff08x-6\uff09\uff08x+24\uff09\uff0e
188\uff0e\uff08x+6\uff09\uff08x+24\uff09\uff0e
189\uff0e\uff08x2-2\uff09\uff08x2+4\uff09\uff0e
190\uff0e3\uff08x2+3\uff09\uff08x+1\uff09\uff08x-1\uff09\uff0e
191\uff0e\uff08m2-17\uff09\uff081+m\uff09\uff081-m\uff09\uff0e
192\uff0e\uff083x2+5y2\uff09\uff08x+2y\uff09\uff08x-2y\uff09\uff0e
193\uff0ex\uff08x+2\uff09\uff08x-2\uff09\uff08x2+2\uff09\uff0e
194\uff0ea\uff08a-20b\uff09\uff08a+15b\uff09\uff0e
195\uff0ex2\uff08x+3\uff09\uff08x2-3x+9\uff09\uff08x-2\uff09\uff08x2+2x+4\uff09\uff0e
\u63d0\u793a\uff1a\u539f\u5f0f=x2\uff08x6+19x3-216\uff09
=x2\uff08x3+27\uff09\uff08x3-8\uff09
=x3\uff08x+3\uff09\uff08x2-3x+9\uff09\uff08x-2\uff09\uff08x2+2x+4\uff09\uff0e
196\uff0eak\uff082a2n-5\uff09\uff083a2n+7\uff09\uff0e
199\uff0e2\uff083x-7y\uff09\uff085x+13y\uff09\uff0e
200\uff0e\uff08m2+15\uff09\uff08m+1\uff09\uff08m-1\uff09\uff0e
201\uff0em2(p\uff0dq)\uff0dp\uff0bq\uff1b
202\uff0ea(ab\uff0bbc\uff0bac)\uff0dabc\uff1b
203\uff0ex4\uff0d2y4\uff0d2x3y\uff0bxy3\uff1b
204\uff0eabc(a2\uff0bb2\uff0bc2)\uff0da3bc\uff0b2ab2c2\uff1b
205\uff0ea2(b\uff0dc)\uff0bb2(c\uff0da)\uff0bc2(a\uff0db)\uff1b
206\uff0e(x2\uff0d2x)2\uff0b2x(x\uff0d2)\uff0b1\uff1b
207\uff0e(x\uff0dy)2\uff0b12(y\uff0dx)z\uff0b36z2\uff1b
208\uff0ex2\uff0d4ax\uff0b8ab\uff0d4b2\uff1b
209\uff0e(ax\uff0bby)2\uff0b(ay\uff0dbx)2\uff0b2(ax\uff0bby)(ay\uff0dbx)\uff1b
210\uff0e(1\uff0da2)(1\uff0db2)\uff0d(a2\uff0d1)2(b2\uff0d1)2\uff1b
211\uff0e(x\uff0b1)2\uff0d9(x\uff0d1)2\uff1b
212\uff0e4a2b2\uff0d(a2\uff0bb2\uff0dc2)2\uff1b
213\uff0eab2\uff0dac2\uff0b4ac\uff0d4a\uff1b
214\uff0ex3n\uff0by3n\uff1b
215\uff0e(x\uff0by)3\uff0b125\uff1b
216\uff0e(3m\uff0d2n)3\uff0b(3m\uff0b2n)3\uff1b
217\uff0ex6(x2\uff0dy2)\uff0by6(y2\uff0dx2)\uff1b
218\uff0e8(x\uff0by)3\uff0b1\uff1b
219\uff0e(a\uff0bb\uff0bc)3\uff0da3\uff0db3\uff0dc3\uff1b
220\uff0ex2\uff0b4xy\uff0b3y2\uff1b
221\uff0ex2\uff0b18x\uff0d144\uff1b
222\uff0ex4\uff0b2x2\uff0d8\uff1b
223\uff0e\uff0dm4\uff0b18m2\uff0d17\uff1b
224\uff0ex5\uff0d2x3\uff0d8x\uff1b
225\uff0ex8\uff0b19x5\uff0d216x2\uff1b
226\uff0e(x2\uff0d7x)2\uff0b10(x2\uff0d7x)\uff0d24\uff1b
227\uff0e5\uff0b7(a\uff0b1)\uff0d6(a\uff0b1)2\uff1b
228\uff0e(x2\uff0bx)(x2\uff0bx\uff0d1)\uff0d2\uff1b
229\uff0ex2\uff0by2\uff0dx2y2\uff0d4xy\uff0d1\uff1b
230\uff0e(x\uff0d1)(x\uff0d2)(x\uff0d3)(x\uff0d4)\uff0d48\uff1b
231\uff0ex2\uff0dy2\uff0dx\uff0dy\uff1b
232\uff0eax2\uff0dbx2\uff0dbx\uff0bax\uff0d3a\uff0b3b\uff1b
233\uff0em4\uff0bm2\uff0b1\uff1b
234\uff0ea2\uff0db2\uff0b2ac\uff0bc2\uff1b
235\uff0ea3\uff0dab2\uff0ba\uff0db\uff1b
236\uff0e625b4\uff0d(a\uff0db)4\uff1b
237\uff0ex6\uff0dy6\uff0b3x2y4\uff0d3x4y2\uff1b
238\uff0ex2\uff0b4xy\uff0b4y2\uff0d2x\uff0d4y\uff0d35\uff1b
239\uff0em2\uff0da2\uff0b4ab\uff0d4b2\uff1b
240\uff0e5m\uff0d5n\uff0dm2\uff0b2mn\uff0dn2\uff0e
\u53c2\u8003\u7b54\u6848:
\u4e09\u3001\u56e0\u5f0f\u5206\u89e3\uff1a
1\uff0e(p\uff0dq)(m\uff0d1)(m\uff0b1)\uff0e
8\uff0e(x\uff0d2b)(x\uff0d4a\uff0b2b)\uff0e
11\uff0e4(2x\uff0d1)(2\uff0dx)\uff0e
20\uff0e(x\uff0b3y)(x\uff0by)\uff0e
21\uff0e(x\uff0d6)(x\uff0b24)\uff0e
27\uff0e(3\uff0b2a)(2\uff0d3a)\uff0e
31\uff0e(x\uff0by)(x\uff0dy\uff0d1)\uff0e
38\uff0e(x\uff0b2y\uff0d7)(x\uff0b2y\uff0b5)\uff0e

1.a^4-4a+3 2.(a+x)^m+1*(b+x)^n-1-(a+x)^m*(b+x)^n 3.x^2+(a+1/a)xy+y^2 4.9a^2-4b^2+4bc-c^2 5.(c-a)^2-4(b-c)(a-b) 答案1.原式=a^4-a-3a+3=(a-1)(a^3+a^2+a-3) 2.[1-(a+x)^m][(b+x)^n-1] 3.(ax+y)(1/ax+y) 4.9a^2-4b^2+4bc-c^2=(3a)^2-(4b^2-4bc+c^2)=(3a)^2-(2b-c)^2=(3a+2b-c)(3a-2b+c) 5.(c-a)^2-4(b-c)(a-b) = (c-a)(c-a)-4(ab-b^2-ac+bc) =c^2-2ac+a^2-4ab+4b^2+4ac-4bc =c^2+a^2+4b^2-4ab+2ac-4bc =(a-2b)^2+c^2-(2c)(a-2b) =(a-2b-c)^2 1.x^2+2x-8 2.x^2+3x-10 3.x^2-x-20 4.x^2+x-6 5.2x^2+5x-3 6.6x^2+4x-2 7.x^2-2x-3 8.x^2+6x+8 9.x^2-x-12 10.x^2-7x+10 11.6x^2+x+2 12.4x^2+4x-3 解方程：（x的平方+5x-6）分之一=（x的平方+x+6）分之一 十字相乘法虽然比较难学,但是一旦学会了它,用它来解题,会给我们带来很多方便,以下是我对十字相乘法提出的一些个人见解。 1、十字相乘法的方法：十字左边相乘等于二次项系数，右边相乘等于常数项，交叉相乘再相加等于一次项系数。 2、十字相乘法的用处：（1）用十字相乘法来分解因式。（2）用十字相乘法来解一元二次方程。 3、十字相乘法的优点：用十字相乘法来解题的速度比较快，能够节约时间，而且运用算量不大，不容易出错。 4、十字相乘法的缺陷：1、有些题目用十字相乘法来解比较简单，但并不是每一道题用十字相乘法来解都简单。2、十字相乘法只适用于二次三项式类型的题目。3、十字相乘法比较难学。 5、十字相乘法解题实例： 1)、 用十字相乘法解一些简单常见的题目 例1把m²+4m-12分解因式 分析：本题中常数项-12可以分为-1×12，-2×6，-3×4，-4×3，-6×2，-12×1当-12分成-2×6时，才符合本题 解：因为 1 -2 1 ╳ 6 所以m²+4m-12=（m-2）（m+6） 例2把5x²+6x-8分解因式 分析：本题中的5可分为1×5,-8可分为-1×8，-2×4，-4×2，-8×1。当二次项系数分为1×5，常数项分为-4×2时，才符合本题 解： 因为 1 2 5 ╳ -4 所以5x²+6x-8=（x+2）（5x-4） 例3解方程x²-8x+15=0 分析：把x²-8x+15看成关于x的一个二次三项式，则15可分成1×15，3×5。 解： 因为 1 -3 1 ╳ -5 所以原方程可变形（x-3）（x-5）=0 所以x1=3 x2=5 例4、解方程 6x²-5x-25=0 分析：把6x²-5x-25看成一个关于x的二次三项式，则6可以分为1×6，2×3，-25可以分成-1×25，-5×5，-25×1。 解： 因为 2 -5 3 ╳ 5 所以 原方程可变形成（2x-5）（3x+5）=0 所以 x1=5/2 x2=-5/3 2)、用十字相乘法解一些比较难的题目 例5把14x²-67xy+18y²分解因式 分析：把14x²-67xy+18y²看成是一个关于x的二次三项式,则14可分为1×14,2×7, 18y²可分为y.18y , 2y.9y , 3y.6y 解: 因为 2 -9y 7 ╳ -2y 所以 14x²-67xy+18y²= (2x-9y)(7x-2y) 例6 把10x²-27xy-28y²-x+25y-3分解因式 分析：在本题中，要把这个多项式整理成二次三项式的形式 解法一、10x²-27xy-28y²-x+25y-3 =10x²-（27y+1）x -（28y²-25y+3） 4y -3 7y ╳ -1 =10x²-（27y+1）x -（4y-3）（7y -1） =[2x -（7y -1）][5x +（4y -3）] 2 -（7y – 1） 5 ╳ 4y - 3 =（2x -7y +1）（5x +4y -3） 说明：在本题中先把28y²-25y+3用十字相乘法分解为（4y-3）（7y -1），再用十字相乘法把10x²-（27y+1）x -（4y-3）（7y -1）分解为[2x -（7y -1）][5x +（4y -3）] 解法二、10x²-27xy-28y²-x+25y-3 =（2x -7y）（5x +4y）-（x -25y）- 3 2 -7y =[（2x -7y）+1] [（5x -4y）-3] 5 ╳ 4y =（2x -7y+1）（5x -4y -3） 2 x -7y 1 5 x - 4y ╳ -3 说明:在本题中先把10x²-27xy-28y²用十字相乘法分解为（2x -7y）（5x +4y）,再把（2x -7y）（5x +4y）-（x -25y）- 3用十字相乘法分解为[（2x -7y）+1] [（5x -4y）-3]. 例7：解关于x方程：x²- 3ax + 2a²–ab -b²=0 分析：2a²–ab-b²可以用十字相乘法进行因式分解 解：x²- 3ax + 2a²–ab -b²=0 x²- 3ax +（2a²–ab - b²）=0 x²- 3ax +（2a+b）（a-b）=0 1 -b 2 ╳ +b [x-（2a+b）][ x-（a-b）]=0 1 -（2a+b） 1 ╳ -（a-b） 所以 x1=2a+b x2=a-b 5-7(a+1)-6(a+1)^2 =-[6(a+1)^2+7(a+1)-5] =-[2(a+1)-1][3(a+1)+5] =-(2a+1)(3a+8); -4x^3 +6x^2 -2x =-2x(2x^2-3x+1) =-2x(x-1)(2x-1); 6(y-z)^2 +13(z-y)+6 =6(z-y)^2+13(z-y)+6 =[2(z-y)+3][3(z-y)+2] =(2z-2y+3)(3z-3y+2). 比如...x^2+6x-7这个式子 由于一次幂x前系数为6 所以，我们可以想到，7-1=6 那正好这个式子的常数项为-7 因此我们想到将-7看成7*（-1） 于是我们作十字相成 x +7 x -1 的到（x+7）·（x-1） 成功分解了因式 3ab^2-9a^2b^2+6a^3b^2 =3ab^2(1-3a+2a^2) =3ab^2(2a^2-3a+1) =3ab^2(2a-1)(a-1) 5-7(a+1)-6(a+1)^2 =-[6(a+1)^2+7(a+1)-5] =-[2(a+1)-1][3(a+1)+5] =-(2a+1)(3a+8); -4x^3 +6x^2 -2x =-2x(2x^2-3x+1) =-2x(x-1)(2x-1); 6(y-z)^2 +13(z-y)+6 =6(z-y)^2+13(z-y)+6 =[2(z-y)+3][3(z-y)+2] =(2z-2y+3)(3z-3y+2). 比如...x^2+6x-7这个式子 由于一次幂x前系数为6 所以，我们可以想到，7-1=6 那正好这个式子的常数项为-7 因此我们想到将-7看成7*（-1） 于是我们作十字相成 x +7 x -1 的到（x+7）·（x-1） 成功分解了因式 3ab^2-9a^2b^2+6a^3b^2 =3ab^2(1-3a+2a^2) =3ab^2(2a^2-3a+1) =3ab^2(2a-1)(a-1) x^2+3x-40 =x^2+3x+2.25-42.25 =(x+1.5)^2-(6.5)^2 =(x+8)(x-5)． ⑹十字相乘法 这种方法有两种情况。 ①x^2+(p+q)x+pq型的式子的因式分解 这类二次三项式的特点是：二次项的系数是1；常数项是两个数的积；一次项系数是常数项的两个因数的和。因此，可以直接将某些二次项的系数是1的二次三项式因式分解：x^2+(p+q)x+pq=(x+p)(x+q) ． ②kx^2+mx+n型的式子的因式分解 如果如果有k=ac，n=bd，且有ad+bc=m时，那么kx^2+mx+n=(ax+b)(cx+d)． 图示如下： a b × c d 例如：因为 1 -3 × 7 2 -3×7=-21，1×2=2，且2-21=-19， 所以7x^2-19x-6=(7x+2)(x-3)． 十字相乘法口诀：首尾分解，交叉相乘，求和凑中 ⑶分组分解法 分组分解是解方程的一种简洁的方法，我们来学习这个知识。 能分组分解的方程有四项或大于四项，一般的分组分解有两种形式：二二分法，三一分法。 比如： ax+ay+bx+by =a(x+y)+b(x+y) =(a+b)(x+y) 我们把ax和ay分一组，bx和by分一组，利用乘法分配律，两两相配，立即解除了困难。 同样，这道题也可以这样做。 ax+ay+bx+by =x(a+b)+y(a+b) =(a+b)(x+y) 几道例题： 1. 5ax+5bx+3ay+3by 解法：=5x(a+b)+3y(a+b) =(5x+3y)(a+b) 说明：系数不一样一样可以做分组分解，和上面一样，把5ax和5bx看成整体，把3ay和3by看成一个整体，利用乘法分配律轻松解出。 2. x3-x2+x-1 解法：=(x3-x2)+(x-1) =x2(x-1)+(x-1) =(x-1)(x2+1) 利用二二分法，提公因式法提出x2，然后相合轻松解决。 3. x2-x-y2-y 解法：=(x2-y2)-(x+y) =(x+y)(x-y)-(x+y) =(x+y)(x-y+1) 利用二二分法，再利用公式法a2-b2=(a+b)(a-b)，然后相合解决。 758²—258² =(758+258)(758-258)=1016*500=508000

鍥犲紡鍒嗚В缁冧範棰
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