经典的因式分解练习题有答案5.docx
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经典的因式分解练习题有答案5
因式分解练习题
一、填空题:
2.(a-3)(3-2a)=_______(3-a)(3-2a);
12.若m2-3m+2=(m+a)(m+b),则a=______,b=______;
15.当m=______时,x2+2(m-3)x+25是完全平方式.
二、选择题:
1.下列各式的因式分解结果中,正确的是()
A.a2b+7ab-b=b(a2+7a)B.3x2y-3xy-6y=3y(x-2)(x+1)
C.8xyz-6x2y2=2xyz(4-3xy)D.-2a2+4ab-6ac=-2a(a+2b-3c)
2.多项式m(n-2)-m2(2-n)分解因式等于()
A.(n-2)(m+m2) B.(n-2)(m-m2)C.m(n-2)(m+1) D.m(n-2)(m-1)
3.在下列等式中,属于因式分解的是()
A.a(x-y)+b(m+n)=ax+bm-ay+bnB.a2-2ab+b2+1=(a-b)2+1
C.-4a2+9b2=(-2a+3b)(2a+3b)D.x2-7x-8=x(x-7)-8
4.下列各式中,能用平方差公式分解因式的是()
A.a2+b2 B.-a2+b2C.-a2-b2 D.-(-a2)+b2
5.若9x2+mxy+16y2是一个完全平方式,那么m的值是()
A.-12 B.±24C.12D.±12
6.把多项式an+4-an+1分解得()
A.an(a4-a)B.an-1(a3-1)C.an+1(a-1)(a2-a+1) D.an+1(a-1)(a2+a+1)
7.若a2+a=-1,则a4+2a3-3a2-4a+3的值为()
A.8 B.7C.10 D.12
8.已知x2+y2+2x-6y+10=0,那么x,y的值分别为()
A.x=1,y=3 B.x=1,y=-3C.x=-1,y=3 D.x=1,y=-3
9.把(m2+3m)4-8(m2+3m)2+16分解因式得()
A.(m+1)4(m+2)2 B.(m-1)2(m-2)2(m2+3m-2)
C.(m+4)2(m-1)2 D.(m+1)2(m+2)2(m2+3m-2)2
10.把x2-7x-60分解因式,得()
A.(x-10)(x+6) B.(x+5)(x-12)C.(x+3)(x-20) D.(x-5)(x+12)
11.把3x2-2xy-8y2分解因式,得()
A.(3x+4)(x-2) B.(3x-4)(x+2)C.(3x+4y)(x-2y) D.(3x-4y)(x+2y)
12.把a2+8ab-33b2分解因式,得()
A.(a+11)(a-3)B.(a-11b)(a-3b)C.(a+11b)(a-3b) D.(a-11b)(a+3b)
13.把x4-3x2+2分解因式,得()
A.(x2-2)(x2-1) B.(x2-2)(x+1)(x-1)
C.(x2+2)(x2+1) D.(x2+2)(x+1)(x-1)
14.多项式x2-ax-bx+ab可分解因式为()
A.-(x+a)(x+b) B.(x-a)(x+b)C.(x-a)(x-b) D.(x+a)(x+b)
15.一个关于x的二次三项式,其x2项的系数是1,常数项是-12,且能分解因式,这样的二次三项式是()
A.x2-11x-12或x2+11x-12B.x2-x-12或x2+x-12
C.x2-4x-12或x2+4x-12D.以上都可以
16.下列各式x3-x2-x+1,x2+y-xy-x,x2-2x-y2+1,(x2+3x)2-(2x+1)2中,不含有(x-1)因式的有()
A.1个 B.2个C.3个 D.4个
17.把9-x2+12xy-36y2分解因式为()
A.(x-6y+3)(x-6x-3)B.-(x-6y+3)(x-6y-3)
C.-(x-6y+3)(x+6y-3)D.-(x-6y+3)(x-6y+3)
18.下列因式分解错误的是()
A.a2-bc+ac-ab=(a-b)(a+c)B.ab-5a+3b-15=(b-5)(a+3)
C.x2+3xy-2x-6y=(x+3y)(x-2)D.x2-6xy-1+9y2=(x+3y+1)(x+3y-1)
19.已知a2x2±2x+b2是完全平方式,且a,b都不为零,则a与b的关系为()
A.互为倒数或互为负倒数 B.互为相反数
C.相等的数 D.任意有理数
20.对x4+4进行因式分解,所得的正确结论是()
A.不能分解因式 B.有因式x2+2x+2C.(xy+2)(xy-8)D.(xy-2)(xy-8)
21.把a4+2a2b2+b4-a2b2分解因式为()
A.(a2+b2+ab)2 B.(a2+b2+ab)(a2+b2-ab)
C.(a2-b2+ab)(a2-b2-ab) D.(a2+b2-ab)2
22.-(3x-1)(x+2y)是下列哪个多项式的分解结果()
A.3x2+6xy-x-2y B.3x2-6xy+x-2y
C.x+2y+3x2+6xy D.x+2y-3x2-6xy
23.64a8-b2因式分解为()
A.(64a4-b)(a4+b) B.(16a2-b)(4a2+b)
C.(8a4-b)(8a4+b) D.(8a2-b)(8a4+b)
24.9(x-y)2+12(x2-y2)+4(x+y)2因式分解为()
A.(5x-y)2 B.(5x+y)2C.(3x-2y)(3x+2y) D.(5x-2y)2
25.(2y-3x)2-2(3x-2y)+1因式分解为()
A.(3x-2y-1)2 B.(3x+2y+1)2
C.(3x-2y+1)2 D.(2y-3x-1)2
26.把(a+b)2-4(a2-b2)+4(a-b)2分解因式为()
A.(3a-b)2 B.(3b+a)2C.(3b-a)2 D.(3a+b)2
27.把a2(b+c)2-2ab(a-c)(b+c)+b2(a-c)2分解因式为()
A.c(a+b)2 B.c(a-b)2C.c2(a+b)2 D.c2(a-b)
28.若4xy-4x2-y2-k有一个因式为(1-2x+y),则k的值为()
A.0 B.1C.-1 D.4
29.分解因式3a2x-4b2y-3b2x+4a2y,正确的是()
A.-(a2+b2)(3x+4y) B.(a-b)(a+b)(3x+4y)
C.(a2+b2)(3x-4y) D.(a-b)(a+b)(3x-4y)
30.分解因式2a2+4ab+2b2-8c2,正确的是()
A.2(a+b-2c) B.2(a+b+c)(a+b-c)
C.(2a+b+4c)(2a+b-4c) D.2(a+b+2c)(a+b-2c)
三、因式分解:
1.m2(p-q)-p+q;2.a(ab+bc+ac)-abc;
3.x4-2y4-2x3y+xy3;4.abc(a2+b2+c2)-a3bc+2ab2c2;
5.a2(b-c)+b2(c-a)+c2(a-b);6.(x2-2x)2+2x(x-2)+1;
7.(x-y)2+12(y-x)z+36z2;8.x2-4ax+8ab-4b2;
9.(ax+by)2+(ay-bx)2+2(ax+by)(ay-bx);10.(1-a2)(1-b2)-(a2-1)2(b2-1)2;
11.(x+1)2-9(x-1)2;12.4a2b2-(a2+b2-c2)2;
13.ab2-ac2+4ac-4a;14.x3n+y3n;
15.(x+y)3+125;16.(3m-2n)3+(3m+2n)3;
17.x6(x2-y2)+y6(y2-x2);18.8(x+y)3+1;
19.(a+b+c)3-a3-b3-c3;20.x2+4xy+3y2;
21.x2+18x-144;22.x4+2x2-8;
23.-m4+18m2-17;24.x5-2x3-8x;
25.x8+19x5-216x2;26.(x2-7x)2+10(x2-7x)-24;
27.5+7(a+1)-6(a+1)2;28.(x2+x)(x2+x-1)-2;
29.x2+y2-x2y2-4xy-1;30.(x-1)(x-2)(x-3)(x-4)-48;
四、证明(求值):
1.已知a+b=0,求a3-2b3+a2b-2ab2的值.
2.求证:
四个连续自然数的积再加上1,一定是一个完全平方数.
3.证明:
(ac-bd)2+(bc+ad)2=(a2+b2)(c2+d2).
4.已知a=k+3,b=2k+2,c=3k-1,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.当a为何值时,多项式x2+7xy+ay2-5x+43y-24可以分解为两个一次因式的乘积.
7.若x,y为任意有理数,比较6xy与x2+9y2的大小.
8.两个连续偶数的平方差是4的倍数.
参考答案:
一、填空题:
7.9,(3a-1)
10.x-5y,x-5y,x-5y,2a-b
11.+5,-2
12.-1,-2(或-2,-1)
14.bc+ac,a+b,a-c
15.8或-2
二、选择题:
1.B 2.C 3.C 4.B 5.B 6.D 7.A 8.C 9.D 10.B 11.C 12.C 13.B 14.C 15.D 16.B 17.B 18.D 19.A 20.B 21.B 22.D 23.C24.A 25.A 26.C 27.C 28.C 29.D 30.D
三、因式分解:
1.(p-q)(m-1)(m+1).
8.(x-2b)(x-4a+2b).
11.4(2x-1)(2-x).
20.(x+3y)(x+y).
21.(x-6)(x+24).
27.(3+2a)(2-3a).
四、证明(求值):
2.提示:
设四个连续自然数为n,n+1,n+2,n+3
6.提示:
a=-18.
∴a=-18.
(1)a的四次方+a²+1
=a^4+2a²+1-a²
=(a²+1)²-a²
=(a²+a+1)(a²-a+1)
(2)2a²-7ab+6b²
=(2a-3b)(a-2b)
(3)3x²+xy-2y²
=(3x-2y)(x+y)
(4)10a²b²+11ab-6
=(2ab+3)(5ab-2)
(5)7a³x-28a(x的五次方)
=7a³(x-4a²)
(6)x³-10x²+16x
=x(x²-10x+16)
=x(x-2)(x-8)
(7)(x²+3x)²-2(x²+3x)-8
=(x²+3x+2)(x²+3x-4)
=(x+1)(x+2)(x+4)(x-1)
(8)(x-y)²+4xy-1
=x²-2xy+y²+4xy-1
=x²+2xy+y²-1
=(x+y)²-1
=(x+y+1)(x+y-1)
9.x^2-4xy+4y^2-x+2y-2
=(x-2y)^2-(x-2y)-2
=(x-2y+1)(x-2y-2)
10.x³+ax²+bx²+cx²+abx+bcx+acx+ab
=x³+cx²+abx+abc+bx²+bcx+ax²+acx
=x²(x+c)+ab(x+c)+bx(x+c)+ax(x+c)
=(x+c)[x²+bx+ab+ax]
=(x+c)[x(x+b)+a(x+b)]
=(x+a)(x+b)(x+c)
11.x²-y²-y-1/2
=x²-(y²+y+1/4)
=x²-(y+1/2)²
=(x+y+1/2)(x-y-1/2)
12.x的四次方+3x³+6x²-4
=x^4+x³+2x³+2x²+4(x²-1)
=x³(x+1)+2x²(x+1)+(4x-4)(x+1)
=(x+1)(x³+2x²+4x-4)
13.x³-3x²+4
=x³+1-3(x²-1)
=(x+1)(x²-x+1)-(3x-3)(x+1)
=(x+1)(x²-4x+4)
=(x+1)(x-2)²
14.32a[(x²+2x)2]-2a
=2a[16(x²+2x)2-1]
=2a(4x^2+8x+1)(4x^2+8x-1)
15.16+8(x²+4x)+(x²+4x)2
=(4+x²+4x)2
=(x²+4x+4)2
=(x+2)4
16.3xny+9x(n-1)y2+xn+1÷4
=x(n-1)(3xy+9y2+x2÷4)
=x^(n-1)(x^2÷4+3xy+9y^2)
=x^(n-1)(x÷2+3y)^2
=(1/4)[x^(n-1)](x+6y)^2
17.a^4+a³+3a-5
=a²*(a²+a)+3a-5
=3a²+3a-5
=3(a²+a)-5
18.1+x+x(x+1)+x(x+1)^2+x(x+1)^3
=(1+x)+x(x+1)+x(x+1)^2+x(x+1)^3
=(1+x)(1+x)+x(x+1)^2+x(x+1)^3
=(1+x)²(1+x)+x(x+1)^3
=(1+x)³(1+x)
=(1+x)^4
19.(x-y)(2x-2y-3)-2
=(x-y)[2(x-y)-3]-2
=2(x-y)^2-3(x-y)-2
=[(x-y)-2][2(x-y)+1]
=(x-y-2)(2x-2y+1).
x^2-3xy-10y^2+x+9y-2=(x-5y+2)(x+2y-1).X^2-y^2+5x+3y+4=(x+y+1)(x-y+4).
xy+y^2+x-y-2=(y+1)(x+y-2).6x^2-7xy-3y^2-xz+7yz-2z^2=(2x-3y+z)(3x+y-2z).
a^2+2b^2+3c^2+3aB+4ac+5bc=(a+b+c)(a+2b+3c)x^2-8x+7=(x-1)(x-7)
X^2+8X+7=(x+1)(x+7)X^2-10X-11=(x-11)(x+1)
X^+3x-18=(x+6)(x-3)x^2+11X+18=(x+2)(x+9)
x^2-11x+18=(x-2)(x-9) x^2+17X-18=(x+18)(x-1)
X^2-17X-18=(x-18)(x+1)xy+6-2x-3y=(x-3)(y-2)
x^2(x-y)+y^2(y-x)=(x+y)(x-y)^22x^2-(a-2b)x-ab=(2x-a)(x+b)
a^4-9a^2b^2=a^2(a+3b)(a-3b) x^3+3x^2-4=(x-1)(x+2)^2
ab(x^2-y^2)+xy(a^2-b^2)=(ay+bx)(ax-by)(x+y)(a-b-c)+(x-y)(b+c-a)=2y(a-b-c)
a^2-a-b^2-b=(a+b)(a-b-1)
(3a-b)^2-4(3a-b)(a+3b)+4(a+3b)^2=[3a-b-2(a+3b)]^2=(a-7b)^2
(x+1)^2(x+2)-(x+1)(x+2)^2=-(x+1)(x+2)abc+ab-4a=a(bc+b-4)
16x^2-81=(4x+9)(4x-9) 9x^2-30x+25=(3x-5)^2
x^2-7x-30=(x-10)(x+3) x^2-25=(x+5)(x-5)
x^2+4x+3=(x+1)(x+3)4x^2-12x+5=(2x-1)(2x-5)
3ax^2-6ax=3ax(x-2) x(x+2)-x=x(x+1)
x^2-4x-ax+4a=(x-4)(x-a) 25x^2-49=(5x-9)(5x+9)
36x^2-60x+25=(6x-5)^24x^2+12x+9=(2x+3)^2
x^2-9x+18=(x-3)(x-6) 2x^2-5x-3=(x-3)(2x+1)
12x^2-50x+8=2(6x-1)(x-4) (x+2)(x-3)+(x+2)(x+4)=(x+2)(2x-1)
2ax^2-3x+2ax-3=(x+1)(2ax-3)9x^2-66x+121=(3x-11)^2
8-2x^2=2(2+x)(2-x) 9x^2-30x+25=(3x-5)^2
-20x^2+9x+20=(-4x+5)(5x+4)12x^2-29x+15=(4x-3)(3x-5)
36x^2+39x+9=3(3x+1)(4x+3)21x^2-31x-22=(21x+11)(x-2)
9x^4-35x^2-4=(9x^2+1)(x+2)(x-2) (2x+1)(x+1)+(2x+1)(x-3)=2(x-1)(2x+1)
2ax^2-3x+2ax-3=(x+1)(2ax-3) x(y+2)-x-y-1=(x-1)(y+1)
(x^2-3x)+(x-3)^2=(x-3)(2x-3) 9x^2-66x+121=(3x-11)^2
8-2x^2=2(2-x)(2+x)x^4-1=(x-1)(x+1)(x^2+1)
x^2+4x-xy-2y+4=(x+2)(x-y+2) 4x^2-12x+5=(2x-1)(2x-5)
21x^2-31x-22=(21x+11)(x-2) 4x^2+4xy+y^2-4x-2y-3=(2x+y-3)(2x+y+1)
9x^5-35x^3-4x=x(9x^2+1)(x+2)(x-2)3x^2-6x=3x(x-2)
49x^2-25=(7x+5)(7x-5)6x^2-13x+5=(2x-1)(3x-5)
x^2+2-3x=(x-1)(x-2) 12x^2-23x-24=(3x-8)(4x+3)
(x+6)(x-6)-(x-6)=(x-6)(x+5) 3(x+2)(x-5)-(x+2)(x-3)=2(x-6)(x+2)
9x2+42x+49=(3x+7)^2.3a3b2c-6a2b2c2+9ab2c3=3ab^2c(a^2-2ac+3c^2)
xy+6-2x-3y=(x-3)(y-2) x2(x-y)+y2(y-x)=(x+y)(x-y)^2
2x2-(a-2b)x-ab=(2x-a)(x+b) x^3+2x^2-16x-32=(x+2)(x+4)(x-4)
(a-b)a^6+(b-a)b^6=(a-b)^2(a^2+ab+b^2)(a+b)(a^2-ab+b^2)(x+y)(x+y+2xy)+(xy+1)(xy-1)=(x+1)(y+1)(x+y+xy-1)
X^8+X^7+1=(X^2+X+1)(X^6-X^4+X^3-X+1) x^8+x^6+x^4+x^2+1=(x^4+x^3-x+1)(x^4+x^3+x^2+x+1)
x^2-20x+100=(x-10)^2x^2-3x+2=(x-1)(x-2)
x^2-2x-15=(x+3)(x-5)x^2+2x-15=(x-3)(x+5)
x^2-7x+12=(x-3)(x-4)x^2-7x-30=(x+3)(x-10)
x^2+2x-3=(x-1)(x+3)x^2-16x+64=(x-8)^2
x^2-9x+20=(x-4)(x-5)x^2-6x+8=(x-2)(x-4)
x^2-6x-7=(x-7)(x+1)