整式的乘法习题精选

整式的乘法习题精选

选择题：

1．下列各式中，正确的是( )

A．t2·t3 = t5 B．t4+t2 = t 6 C．t3·t4 = t12 D．t5·t5 = 2t5

答案：A

说明：t4与t2不是同类项，不能合并，B错；同底数幂相乘，底不变，指数相加，所以t3·t4 = t3+4 = t7≠t12，C错；t5•t5 = t5+5 = t10≠2t5，D错；t2•t3 = t2+3 = t5，A正确；答案为A．

2．下列计算错误的是( )

A．−a2·(−a)2 = −a4 B．(−a)2·(−a)4 = a6

C．(−a3)·(−a)2 = a5 D．(−a)·(−a)2 = −a3

答案：C

说明：−a2·(−a)2 = −a2·a2 = −a2+2 = −a4，A计算正确；(−a)2·(−a)4 = a2·a4 = a2+4 = a6，B计算正确；(−a3)·(−a)2 = −a3·a2 = −a5≠a5，C计算错误；(−a)·(−a)2 = −a·a2 = −a3，D计算正确；所以答案为C．

3．下列计算中，运算正确的个数是( )

①5x3−x3 = x3 ② 3m·2n = 6m+n

③am+an = am+n ④xm+1·xm+2 = xm·xm+3

A．1 B． 2 C．3 D．4

答案：A

说明：5x3−x3 = (5−1)x3 = 4x3 ≠x3 ，①错误； 3m与2n 不是同底数幂，它们相乘把底数相乘而指数相加显然是不对的，比如m = 1，n = 2，则 3m·2n = 31·22 = 3·4 = 12，而 6m+n = 61+2 = 63 = 216≠12，②错误；am与an只有在m = n时才是同类项，此时am+an = 2am≠am+n，而在m≠n时，am与an无法合并，③错；xm+1·xm+2 = xm+1+m+2 = xm+m+3 = xm·xm+3，④正确；所以答案为A．

4．计算a6(a2)3的结果等于( )

A．a11 B．a 12 C．a14 D．a36

答案：B

说明：a6(a2)3 = a6·a2×3 = a6·a6 = a6+6 = a12，所以答案为B．

5．下列各式计算中，正确的是( )

A．(a3)3 = a6 B．(−a5)4 = −a 20 C．[(−a)5]3 = a15 D．[(−a)2]3 = a6

答案：D

说明：(a3)3 = a3×3 = a9，A错；(−a5)4 = a5×4 = a20，B错；[(−a)5]3 = (−a)5×3 = (−a)15 = −a15，C错；[(−a)2]3 = (−a)2×3 = (−a)6 = a6，D正确，答案为D．

6．下列各式计算中，错误的是( )

A．(m6)6 = m36 B．(a4)m = (a 2m) 2 C．x2n = (−xn)2 D．x2n = (−x2)n

答案：D

说明：(m6)6 = m6×6 = m36，A计算正确；(a4)m = a 4m，(a 2m)2 = a 4m，B计算正确；(−xn)2 = x2n，C计算正确；当n为偶数时，(−x2)n = (x2)n = x2n；当n为奇数时，(−x2)n = −x2n，所以D不正确，答案为D．

7．下列计算正确的是( )

A．(xy)3 = xy3 B．(2xy)3 = 6x3y3

C．(−3x2)3 = 27x5 D．(a2b)n = a2nbn

答案：D

说明：(xy)3 = x3y3，A错；(2xy)3 = 23x3y3 = 8x3y3，B错；(−3x2)3 = (−3)3(x2)3 = −27x6，

C错；(a2b)n = (a2)nbn = a2nbn，D正确，答案为D．

8．下列各式错误的是( )

A．(23)4 = 212 B．(− 2a)3 = − 8a3

C．(2mn2)4 = 16m4n8 D．(3ab)2 = 6a2b2

答案：C

说明：(23)4 = 23×4 = 212，A中式子正确；(− 2a)3 = (−2) 3a3 = − 8a3，B中式子正确；(3ab)2 = 32a2b2 = 9a2b2，C中式子错误；(2mn2)4 = 24m4(n2)4 = 16m4n8，D中式子正确，所以答案为C．

9．下列计算中，错误的是( )

A．mn·m2n+1 = m3n+1 B．(−am−1)2 = a 2m−2

C．(a2b)n = a2nbn D．(−3x2)3 = −9x6

答案：D

说明：mn·m2n+1 = mn+2n+1 = m3n+1，A中计算正确；(−am−1)2 = a2(m−1) = a 2m−2，B中计算正确； (a2b)n = (a2)nbn = a2nbn，C中计算正确；(−3x2)3 = (−3)3(x2)3 = −27x6，D中计算错误；所以答案为D．

10．下列计算中，错误的是( )

A．(−2ab2)2·(− 3a2b)3 = − 108a8b7

B．(2xy)3·(−2xy)2 = 32x5y5

C．(m2n)(−mn2)2 =m4n4

D．(−xy)2(x2y) = x4y3

答案：C

说明：(−2ab2)2·(− 3a2b)3 = (−2) 2a2(b2)2·(−3)3(a2)3b3 = 4a2b4·(−27)a6b3 = − 108a2+6b4+3 = − 108a8b7，A中计算正确；(2xy)3·(−2xy)2 = (2xy)3·(2xy)2 = (2xy)3+2 = (2xy)5 = 25x5y5 = 32x5y5，B中计算正确；(m2n)(−mn2)2 =m2n(−) 2m2(n2)2 =m2n·m2n4 =

m2+2n1+4 =

m4n5，C中计算错误；(−xy)2(x2y) = (−)2x2y2·x2y

=x2y2·x2y = x4y3，D中计算正确，所以答案为C．

11．下列计算结果正确的是( )

A．(6ab2− 4a2b)•3ab = 18ab2− 12a2b

B．(−x)(2x+x2−1) = −x3−2x2+1

C．(−3x2y)(−2xy+3yz−1) = 6x3y2−9x2y2z2+3x2y

D．(a3−b)•2ab =a4b−ab2

答案：D

说明：(6ab2− 4a2b)•3ab = 6ab2·3ab− 4a2b·3ab = 18a2b3− 12a3b，A计算错误；(−x)(2x+x2−1) = −x·2x+(−x)·x2−(−x) = −2x2−x3+x = −x3−2x2+x，B计算错误；(−3x2y)(−2xy+3yz−1) = (−3x2y) • (−2xy)+(−3x2y) •3yz−(−3x2y) = 6x3y2−9x2y2z+3x2y，C计算错误；(a3−b)•2ab = (a3) •2ab−(b)•2ab =a4b−ab2，D计算正确，所以答案为D．

12．若(x−2)(x+3) = x2+a+b，则a、b的值为( )

A．a = 5，b = 6 B．a = 1，b = −6

C．a = 1，b = 6 D．a = 5，b = −6

答案：B

说明：因为(x−2)(x+3) = x•x−2x+3x−6 = x2+x−6，所以a = 1，b = −6，答案为B．

解答题：

1．计算

(1)(− 5a3b2)·(−3ab 2c)·(− 7a2b)；

(2)− 2a2b3·(m−n)5·ab2·(n−m)2+a2(m−n)·6ab2；

(3) 3a2( ab2−b)−( 2a2b2−3ab)(− 3a)；

(4)(3x2−5y)(x2+2x−3)．

解：(1)(− 5a3b2)·(−3ab 2c)·(− 7a2b) = [(−5)×(−3)×(−7)](a3·a·a2)(b2·b2·b)c = − 105a6b 5c．

(2)− 2a2b3·(m−n)5·ab2·(n−m)2+a2(m−n)·6ab2

= (−2·)·(a2·a)·(b3·b2)[(m−n)5·(m−n)2]+(·6)(a2·a)(m−n)b2 = −a3b5(m−n)7+ 2a3b2(m−n)．

(3) 3a2( ab2−b)−( 2a2b2−3ab)(− 3a) = 3a2·ab2− 3a2b+ 2a2b2· 3a−3ab· 3a

= a3b2− 3a2b+ 6a3b2− 9a2b = 7a3b2− 12a2b．

(4)(3x2−5y)(x2+2x−3) = 3x2·x2−5y·x2+3x2·2x−5y·2x+3x2·(−3)−5y·(−3)

= 3x4−5x2y+6x3−10xy−9x2+15y

= 3x4+6x3−5x2y−9x2−10xy+15y．

2．当x = −3时，求8x2−(x−2)(x+1)−3(x−1)(x−2)的值．

解：8x2−(x−2)(x+1)−3(x−1)(x−2) = 8x2−(x2−2x+x−2)−3(x2−x−2x+2)

= 8x2−x2+x+2−3x2+9x−6 = 4x2+10x−4．

当x = −3时，原式 = 4·(−3)2+10·(−3)−4 = 36−30−4 = 2．

3．把一个长方形的长减少3，宽增加2，面积不变，若长增加1，宽减少1，则面积减少6，求长方形的面积．

解：设长方形的长为x，宽为y，则由题意有

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