《2025年同步测控优化设计八年级数学上册人教版》

典例2 把下列各式分解因式：
(1)$(x - y)^{3} - 3z(y - x)^{2}$；
(2)$a(a - b - c) + b(b + c - a) + c(c - a + b)$；
(3)$m(5ax + ay - 1) - m(3ax - ay - 1)$.

答案：
(1)
$\begin{aligned}&(x - y)^{3} - 3z(y - x)^{2}\\=&(x - y)^{3} - 3z(x - y)^{2}\\=&(x - y)^{2}(x - y - 3z)\end{aligned}$
(2)
$\begin{aligned}&a(a - b - c) + b(b + c - a) + c(c - a + b)\\=&a(a - b - c)-b(a - b - c)-c(a - b - c)\\=&(a - b - c)(a - b - c)\\=&(a - b - c)^{2}\end{aligned}$
(3)
$\begin{aligned}&m(5ax + ay - 1) - m(3ax - ay - 1)\\=&m[(5ax + ay - 1)-(3ax - ay - 1)]\\=&m(5ax + ay - 1 - 3ax + ay + 1)\\=&m(2ax + 2ay)\\=&2m(ax + ay)\\=&2ma(x + y)\end{aligned}$

举一反三 把下列各式分解因式：
(1)$2(a - 3)^{2} - a + 3$；
(2)$3m(x - y) - 2(y - x)^{2}$；
(3)$18b(a - b)^{2} - 12(a - b)^{3}$.

答案： 解（1）原式$=2(a-3)^{2}-(a-3)=(a-3)(2a-7).$
（2）原式$=3m(x-y)-2(x-y)^{2}=(x-y)(3m-2x+2y).$
（3）原式$=6(a-b)^{2}(5b-2a).$