## Putting It Together: Factoring

Factoring polynomials may seem like a tedious task that is put forth only to cause you frustration. Hopefully, now that you have worked through the exercises, text, and videos in this module, you are able to see that it can be a useful tool for solving polynomial equations.

Let us revisit the scenario we put forth in the beginning of the module. Recall that Joan proposed a polynomial function to represent the number of likes or comments received on Instagram based on the number of posts, $x$, in a day.

$L(x)=-x^2+4x$

Also recall that she tried to find the number of posts that would generate no likes or comments for her friend, Nate, by solving the function for zero:

$0=-x^2+4x$

She tried guessing, but after attending office hours and asking her teacher how to solve an equations like this, she knows a more elegant and efficient way to solve the equation. Hopefully she can convince Nate with math that there is a limit to how much people enjoy his posts on Instagram.

First, Joan factors out $-x$

$\begin{array}{c}0=-x^2+4x\\0=-x(x-4)\end{array}$

Next, she can use the principle of zero products to solve for $x$.

$\begin{array}{c}0=-x(x-4)\\0=-x\text{ OR }0=x-4\\0=x\text{ OR }4=x\end{array}$

Factoring has led Joan to a solution much faster than guessing would have. The solution $x=0$ is obvious; if Nate does not post, nobody can like or comment on his posts. The solution $x=4$ can be interpreted to mean that once Nate makes four posts on Instagram in one day, people will stop paying attention.

Optimizing the amount of social media posting you do can influence how widely your ideas are shared. If you use social media for business, it is worth researching how often and when to post so that you will not only gain the audience you want, but also keep their attention.

Factoring may seem meaningless without the context of solving polynomial equations. We hope you will see that it can actually be kind of like solving a fun puzzle, too. Taking the hard route of guessing, much like free climbing a building, may seem fun at first. . . but, like using an elevator, once you discover how to factor you will be able to get where you are going with much less effort (less sweat, too).