###### Example7.2.4

Factor $$t^3-5t^2-3t+15\text{.}$$ This example has a complication with negative signs. If we try to break up this polynomial into two groups as $$\left(t^3-5t^2\right)-(3t+15)\text{,}$$ then we've made an error! In that last expression, we are subtracting a group with the term $$15\text{,}$$ so overall it subtracts $$15\text{.}$$ The original polynomial added $$15\text{,}$$ so we are off course.

One way to handle this is to treat subtraction as addition of a negative:

\begin{align*} t^3-5t^2-3t+15\amp=t^3-5t^2+(-3t)+15\\ \amp=\left(t^3-5t^2\right)+\left(-3t+15\right)\\ \end{align*}

Now we can proceed to factor out common factors from each group. Since the second group leads with a negative coefficient, we'll factor out $$-3\text{.}$$ This will result in the “$${}+15$$” becoming “$${}-5\text{.}$$”

\begin{align*} \amp=\highlight{t^2}(t-5)+\highlight{(-3)}(t-5)\\ \amp=\highlight{t^2}\attention{\overbrace{(t-5)}}\highlight{{}-3}\attention{\overbrace{(t-5)}}\\ \amp=(t-5)\highlight{\left(t^2-3\right)} \end{align*}

And remember that we can confirm this is correct by multiplying it out. If we made no mistakes, it should result in the original $$t^3-5t^2-3t+15\text{.}$$

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