问题 计算：
(1) $(4x^{2}y - x^{3}y^{3}) ÷ (-2x^{2}y)$； (2) $(27a^{3} - 15a^{2} + 3a) ÷ 3a$；
(3) $[(x^{2} + 1)(x^{2} - 1) + 1] ÷ x^{3}$.
名师指导
(1) 注意按照法则进行计算,特别要注意符号问题；
(2) 注意 $3a ÷ 3a = 1$,不要误认为是 $0$；
(3) 注意先算括号内的.
解题示范 (学生在教师指导下,独立完成)
解：
(1) $(4x^{2}y - x^{3}y^{3}) ÷ (-2x^{2}y)$； (2) $(27a^{3} - 15a^{2} + 3a) ÷ 3a$；
(3) $[(x^{2} + 1)(x^{2} - 1) + 1] ÷ x^{3}$.
名师指导
(1) 注意按照法则进行计算,特别要注意符号问题；
(2) 注意 $3a ÷ 3a = 1$,不要误认为是 $0$；
(3) 注意先算括号内的.
解题示范 (学生在教师指导下,独立完成)
解：
答案：(1)
$\begin{aligned}&(4x^{2}y - x^{3}y^{3}) ÷ (-2x^{2}y)\\=&4x^{2}y÷(-2x^{2}y) - x^{3}y^{3}÷(-2x^{2}y)\\=&-2 + \frac{1}{2}xy^{2}\end{aligned}$
(2)
$\begin{aligned}&(27a^{3} - 15a^{2} + 3a) ÷ 3a\\=&27a^{3}÷3a - 15a^{2}÷3a + 3a÷3a\\=&9a^{2} - 5a + 1\end{aligned}$
(3)
$\begin{aligned}&[(x^{2} + 1)(x^{2} - 1) + 1] ÷ x^{3}\\=&[(x^{2})^{2} - 1^{2} + 1] ÷ x^{3}\\=&(x^{4} - 1 + 1) ÷ x^{3}\\=&x^{4}÷x^{3}\\=&x\end{aligned}$
$\begin{aligned}&(4x^{2}y - x^{3}y^{3}) ÷ (-2x^{2}y)\\=&4x^{2}y÷(-2x^{2}y) - x^{3}y^{3}÷(-2x^{2}y)\\=&-2 + \frac{1}{2}xy^{2}\end{aligned}$
(2)
$\begin{aligned}&(27a^{3} - 15a^{2} + 3a) ÷ 3a\\=&27a^{3}÷3a - 15a^{2}÷3a + 3a÷3a\\=&9a^{2} - 5a + 1\end{aligned}$
(3)
$\begin{aligned}&[(x^{2} + 1)(x^{2} - 1) + 1] ÷ x^{3}\\=&[(x^{2})^{2} - 1^{2} + 1] ÷ x^{3}\\=&(x^{4} - 1 + 1) ÷ x^{3}\\=&x^{4}÷x^{3}\\=&x\end{aligned}$
1. 下列运算中正确的是 (
A.$(x^{2}y^{3})^{4} = x^{6}y^{7}$
B.$x^{3} \cdot x^{4} = x^{7}$
C.$(x^{2}y^{3}) ÷ (xy^{3}) = xy$
D.$(21x^{3} - 14x^{2} + 7x) ÷ 7x = 3x^{2} - 2x$
B
)A.$(x^{2}y^{3})^{4} = x^{6}y^{7}$
B.$x^{3} \cdot x^{4} = x^{7}$
C.$(x^{2}y^{3}) ÷ (xy^{3}) = xy$
D.$(21x^{3} - 14x^{2} + 7x) ÷ 7x = 3x^{2} - 2x$
答案：B
解析：
A.$(x^{2}y^{3})^{4}=x^{8}y^{12}$，错误；
B.$x^{3}\cdot x^{4}=x^{7}$，正确；
C.$(x^{2}y^{3})÷(xy^{3})=x$，错误；
D.$(21x^{3}-14x^{2}+7x)÷7x=3x^{2}-2x+1$，错误。
结论：B
B.$x^{3}\cdot x^{4}=x^{7}$，正确；
C.$(x^{2}y^{3})÷(xy^{3})=x$，错误；
D.$(21x^{3}-14x^{2}+7x)÷7x=3x^{2}-2x+1$，错误。
结论：B
2. 已知 $6a^{4}b^{m} ÷ 8a^{n}b = \frac{3}{4}b^{3}$,那么 $m$ 和 $n$ 的值分别为 (
A.$m = 3$,$n = 4$
B.$m = 4$,$n = 3$
C.$m = 3$,$n = 3$
D.$m = 4$,$n = 4$
D
)A.$m = 3$,$n = 4$
B.$m = 4$,$n = 3$
C.$m = 3$,$n = 3$
D.$m = 4$,$n = 4$
答案：D
解析：
$6a^{4}b^{m} ÷ 8a^{n}b = \frac{6}{8}a^{4 - n}b^{m - 1} = \frac{3}{4}a^{4 - n}b^{m - 1}$，
因为$6a^{4}b^{m} ÷ 8a^{n}b = \frac{3}{4}b^{3}$，
所以$\frac{3}{4}a^{4 - n}b^{m - 1} = \frac{3}{4}b^{3}$，
则$4 - n = 0$，$m - 1 = 3$，
解得$n = 4$，$m = 4$。
D
因为$6a^{4}b^{m} ÷ 8a^{n}b = \frac{3}{4}b^{3}$，
所以$\frac{3}{4}a^{4 - n}b^{m - 1} = \frac{3}{4}b^{3}$，
则$4 - n = 0$，$m - 1 = 3$，
解得$n = 4$，$m = 4$。
D
3. 设 $M$ 是一个多项式,且 $M ÷ \frac{5}{3}x^{2}y = -2x^{2}y^{4} + \frac{3}{2}x$,那么 $M$ 等于 (
A.$-\frac{6}{5}x^{4}y^{5} + \frac{9}{10}x^{4}y^{3}$
B.$-\frac{6}{5}y^{3} + \frac{5}{2}xy$
C.$-\frac{10}{3}x^{4}y^{5} + \frac{5}{2}x^{3}y$
D.$\frac{10}{3}x^{4}y^{5} - \frac{5}{2}x^{3}y$
C
)A.$-\frac{6}{5}x^{4}y^{5} + \frac{9}{10}x^{4}y^{3}$
B.$-\frac{6}{5}y^{3} + \frac{5}{2}xy$
C.$-\frac{10}{3}x^{4}y^{5} + \frac{5}{2}x^{3}y$
D.$\frac{10}{3}x^{4}y^{5} - \frac{5}{2}x^{3}y$
答案：C
解析：
$M=\left(-2x^{2}y^{4}+\frac{3}{2}x\right)×\frac{5}{3}x^{2}y$
$=-2x^{2}y^{4}×\frac{5}{3}x^{2}y+\frac{3}{2}x×\frac{5}{3}x^{2}y$
$=-\frac{10}{3}x^{4}y^{5}+\frac{5}{2}x^{3}y$
C
$=-2x^{2}y^{4}×\frac{5}{3}x^{2}y+\frac{3}{2}x×\frac{5}{3}x^{2}y$
$=-\frac{10}{3}x^{4}y^{5}+\frac{5}{2}x^{3}y$
C
4. 下列计算中,错误的是 (
A.$(8x^{3}y - 2x^{2}) ÷ (-2x^{2}) = -4xy + 1$
B.$(\frac{6}{5}a^{3}x^{4} - \frac{9}{10}ax^{5}) ÷ \frac{3}{5}ax^{3} = \frac{1}{2}a^{2}x - \frac{2}{3}x^{2}$
C.$(x^{2n + 1} - x^{2n}) ÷ x^{2n} = x - 1$
D.$(3a^{3} + 2a^{2}) ÷ a^{2} = 3a + 2$
B
)A.$(8x^{3}y - 2x^{2}) ÷ (-2x^{2}) = -4xy + 1$
B.$(\frac{6}{5}a^{3}x^{4} - \frac{9}{10}ax^{5}) ÷ \frac{3}{5}ax^{3} = \frac{1}{2}a^{2}x - \frac{2}{3}x^{2}$
C.$(x^{2n + 1} - x^{2n}) ÷ x^{2n} = x - 1$
D.$(3a^{3} + 2a^{2}) ÷ a^{2} = 3a + 2$
答案：B
解析：
A. $(8x^{3}y - 2x^{2}) ÷ (-2x^{2}) = 8x^{3}y ÷ (-2x^{2}) - 2x^{2} ÷ (-2x^{2}) = -4xy + 1$，正确。
B. $(\frac{6}{5}a^{3}x^{4} - \frac{9}{10}ax^{5}) ÷ \frac{3}{5}ax^{3} = \frac{6}{5}a^{3}x^{4} ÷ \frac{3}{5}ax^{3} - \frac{9}{10}ax^{5} ÷ \frac{3}{5}ax^{3} = 2a^{2}x - \frac{3}{2}x^{2}$，错误。
C. $(x^{2n + 1} - x^{2n}) ÷ x^{2n} = x^{2n + 1} ÷ x^{2n} - x^{2n} ÷ x^{2n} = x - 1$，正确。
D. $(3a^{3} + 2a^{2}) ÷ a^{2} = 3a^{3} ÷ a^{2} + 2a^{2} ÷ a^{2} = 3a + 2$，正确。
B
B. $(\frac{6}{5}a^{3}x^{4} - \frac{9}{10}ax^{5}) ÷ \frac{3}{5}ax^{3} = \frac{6}{5}a^{3}x^{4} ÷ \frac{3}{5}ax^{3} - \frac{9}{10}ax^{5} ÷ \frac{3}{5}ax^{3} = 2a^{2}x - \frac{3}{2}x^{2}$，错误。
C. $(x^{2n + 1} - x^{2n}) ÷ x^{2n} = x^{2n + 1} ÷ x^{2n} - x^{2n} ÷ x^{2n} = x - 1$，正确。
D. $(3a^{3} + 2a^{2}) ÷ a^{2} = 3a^{3} ÷ a^{2} + 2a^{2} ÷ a^{2} = 3a + 2$，正确。
B