經(jīng)典因式分解練習題集(附答案解析)

1、因式分解練習題一、填空題：1. 4a + 8a2 + 24a = 4a()；2. (a 3)(3 2a)=(3- a)(3- 2a);工 a3b_ ab3 = ab(a_b)f ；4 .(1 - a)mn+ a- 1 = ( )(mn- 1);5 . 0 0009x*=(6 1-( )+=仁一)S7. ( )a3 - 6a+ 1 = ( )2;8. 8x3 - ( ) = (2x-)(+ 6x + 9)；9. x3 -y2 -z2 + 2yz = x3 - ( ) = ( X10. 2ax_ 10ay+ 5by _bx= 2a( ) -b()=()()；IL x2 + 3x- 10 = (x

2、)(x)；12. 若 m2 3m+2=(m + a)(m+ b),貝U a=, b=s 1 3 113. x - -y =(x- -y)()； Q414. a2 _be + ab_ ac = (a2 + ab)()=()();15. 當m=時，x2+2(m 3)x+ 25是完全平方式.三、因式分解：1. m2(p-q)-p + q;2. a(ab+ bc+ ac) abc;3. x4 2夕一2x3y+ xy3;4. abc(3+ b2 + c2) a3bc+2at2c2;5. a2(b c)+b2(c a)+ c2(a b);6. (x2 2x)2 + 2x(x 2)+ 1;7. (x y)2

3、+12(y x)z+36z2;8. x2 4ax+ 8ab 4b2;9. (ax+ by)2+ (ay- bx)2+ 2(a奸 by)(ay- bx);10. (1-a2)(1-b2)-(a2-1)2(b2-1)2;11. (x+1)2 9(x 1)2;12. 4s2b2(a2+ b2 c2)2;13. ab2ac2+ 4ac 4a;14. x3n + y3n;18. 8(x+ y)3+1；15. (x + y)3+125;16. (3m 2n)3+(3m+2n；17. x6(x2 y2) + y6(y2 x2)；19. (a+ b+c)3 a3b3 c?;20. x2+4xy+ 3y2;21

4、. x2+ 18x 144;23. m4+18m217;25. x8+ 19x5 216$;27. 5+7(a+ 1)-6(a+ 1)2;29. x2+ y2-x2y2-4xy- 1;30.四、證明(求值)：1 .已知 a+ b=0,求 a32b3 + a2b2ab2的值.2 .求證：四個連續(xù)自然數(shù)的積再加上1,22. x4+2x2-8;24. x5-2x3-8x;26. (x2-7x)2+10(x2-7x)-24;28. (x2+x)(x2 + x-1)-2;(x 1)(x 2)(x 3)(x 4)- 48;是一個完全平方數(shù).3 .證明：(ac bd)2+(bc+ ad)2=(a2+b2)(

5、c2+d2).4 .已知 a=k+3, b=2k + 2, c=3k1,求 a2+b2 + c2+2ab 2bc 2ac的值.5 .若 x2+mx+n=(x 3)(x+ 4),求(m+n)2的值.6 .當a為何值時，多項式x2+7xy+ a9 5x+43y 24可以分解為兩個一次因式的 乘積.7 .若x, y為任意有理數(shù)，比較6xy與x2+ 9y2的大小.8 .兩個連續(xù)偶數(shù)的平方差是4的倍數(shù).參考答案:一、填空題:27, 3, d(y-z)3, x-y+zx + y-zL - + 2a + 64 1-a:二) 0 E：； 7. 9, (3a-1)g3. d+ b a 1 廿 12410. x5

6、y, x5y, x5y, 2a b11. +5, 212. 1, 2(或一2, 1):一.1 .2414. bc+ ac, a+ b, a c15. 8 或一2三、因式分解:1. (pq)(m1)(m+1). 一：3,原式=+ y，-+ y3) = (x3 + y3)(x- 2y) = (x- 2y)(x + y)(Y-xy + /.4. abc(b+c)L5,原式=/b - aac+bac aba + ca (a -b) = (aab-aba) -(aac -bac) + c2 (a - b) = ab(a -b) - c(a2 -b2) + c2 (a - b) = (a - b)ab -

7、 c(a+ b) + c2 =(a - b)a(b - c) - c(t - c) = (a _ b)(b - c)(a - c).6.原式=x(k - 2)F + 2x(x -2) +12 =x(x - 2) + 1 = (x -2x+ I)2 =(籃 Fl7. (x-y-6z)2T8. (x- 2b)(x-4a+ 2b).9. (ax + by+ay-bx).10. (1-a)(l +a)(l-b)(l + b)(r + b-a2b3).11. 4(2x 1)(2 x).12 .原式=(2ab+aa +b3 - ca)(2ab- aa -ba + c2) =(a+ b)a - c3 cJ

8、- (a - b)2 = (a + b+ c)(a+ b - c)(c + a -b)(c - a + b5.13 .原式=a(b3 -c2 + 4c - 4) = a(b2 - c2 + 2b - 2b+ 2c+ 2c 7) =a(b - c)(b + c) + 2(b+ c) - 2(b - c) - 4 = a(b - c) + 2(b + c) - 2 =a(b - c + 2)(b + c - 2).14 .+ 丫以區(qū)a-Fy +產(chǎn)).15 . (x + y+5)(xa + 2xy + y3 -5x- 5y+ 25).16 . 18m(3m2 + 4na),17 .原式-/尸儀+y)

9、僅力(/+產(chǎn)-力二 (x+y)a(x _y)a(x2 -可+為僅+ 取+也.18 . (2x + 2y + l)(4x2 + 8xy + 4y2 - 2x - 2y + 1).19- 3(b+ c)(a + b)(c + a).提示：原式=(a + b + c)，20- (x + 3y)(x+ y).21. (x-6)(x+ 24).22. -2)(/+ 4).23. - (m2 - 17)(m+ l)(m - 1).24. x(x + 2)(x - 2)(/ + 2) .25.原式=x3(xe +19- 216) = xa(x3 + 27)(xi -8) =xx+ 3) (xa - 3x+

10、9)(x -2)(父 + 2x+ 4).26, (x-3)(x-4)(x-7x-2).28,原式=(x3 + x) (x3 +x) -1 2 = (x3 +x)3 -(x3 + x) - 2 = 27. (3+2a)(2 3a).(x3 +x-2)(xa + x+ 1) =(i+ 2)(x -l)(xa +x+l).29.原式= (x3 -2xy + y，Yy, + 2穹+ 】)二(x -y) -(xy+ I)2 = (x - y + xy+ l)(x - y - xy - 1).30,原式=(x - 1)(x-4)(x 2),一3)-犯=(1 一的+ 4(-欠) + 6-48 = (xa-5

11、k)3 + 10(xa-&)-24 = (xa-5x +12)(7 - -2).四、證明(求值)：1,原式二(a3+aJb)(2b3 + 2aba) =aa(a+b) -2ba(a+b)=0.2 .提示：設(shè)四個連續(xù)自然數(shù)為 n, n+1, n+2, n + 3n(n + 1)(口 + 2)( 口 + 3) + 1= (n2 + 3n)(n2 + 3n+ 2) + 1 = (n2 + 3n) + 2(n2+ 3n) + l = (n2 + 3n+ I)3 .3 .證明士(就-bdy + (bc+皆d)，=員- 2貧bcd + b4 + bY + 2abcd+ a2d2 = a2(c2 + d2).4 .提不；aa +ba + c3 + 2ab 2bc _ 2ac = (a + b)3 - 2c(a+b) + ca =(a + b - c)3 = (k+ 3 + 2k. + 2 + 3k+ I)。= 36.5,提不：m = 1, n = T2； (m+療= 121.6.提示:a= -18.令/ + 7xy+ay2 -5x+43y - 24 = (x + my+ n)(x + py+q) = / +(in + p)xy + mpy2 +