###### Example7.1.10

Factor $$-35m^5+5m^4-10m^3\text{.}$$

First, we identify the common factor. The number $$5$$ is the greatest common factor of the three coefficients (which were $$-35\text{,}$$ $$5\text{,}$$ and $$-10$$) and also $$m^3$$ is the largest expression that divides $$m^5\text{,}$$ $$m^4\text{,}$$ and $$m^3\text{.}$$ Therefore the greatest common factor is $$5m^3\text{.}$$

In this example, the leading term is a negative number. When this happens, we will make it common practice to take that negative as part of the greatest common factor. So we will proceed by factoring out $$-5m^3\text{.}$$ Note the sign changes.

\begin{align*} -35m^5\highlight{{}+{}}5m^4\highlight{{}-{}}10m^3\amp=-5m^3(\phantom{7m^2}\highlight{{}-{}}\phantom{m}\highlight{{}+{}}\phantom{2})\\ \amp=-5m^3(7m^2-\phantom{m}+\phantom{2})\\ \amp=-5m^3(7m^2-m+\phantom{2})\\ \amp=-5m^3(7m^2-m+2) \end{align*}
in-context