Example 7.1.10

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*}