## Why It Matters: Factoring

### Why Learn How to Factor?

There are hard ways to do things and there are easy ways to do things. For example, you can free climb the brick wall of your apartment building and crawl into the third floor window, or you could take the stairs. Easier still, you could use the elevator. Math also offers efficient ways to accomplish a task.

Remember Joan? Joan has a friend, Nate, who is never far from his phone or a computer. Nate posts numerous pictures, videos, memes, quips, and links on his various social media accounts daily. Joan follows Nate on Instagram and has lately been annoyed with the sheer number of his posts. She wants to let him know that he is probably annoying others, too, but she does not want to hurt his feelings.

Since Joan is studying polynomial functions in her math class, she comes up with a plan. She will appeal to Nate’s data-savvy, technical side by proposing that you can fall out of favor with your Instagram followers by posting too many times in one day.

She sees the following polynomial function in her math homework:

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

In this function, Joan decides that x represents the number of pictures posted on Instagram each day by her friend Nate. $L(x)$, then, represents the number of likes or comments his pictures on Instagram receive based on the number posted. If he does not post any, obviously no one will like his pictures.

If Nate posts too many, people will get bored of his spam and ignore him. Joan wants to know how much is too much so Nate can stay popular on Instagram and not become an annoying spammer. By solving this polynomial function for zero, Joan hopes to find out exactly how many is too many posts for Nate. This translates to, when will he get zero likes or comments on his Instagram pictures?

Joan is not sure how to solve this kind of equation, so she starts guessing values for x.

Her first guess is $10$ since Nate posts more than that.

$L(10)=-10^2+4(10)=-100+40=-60$

Whoa, if Nate posts $10$ pictures on Instagram each day, he will get -60 likes or comments. Joan assumes negative numbers are not a good thing.

Next, she guesses $5$.

$L(5)=-5^2+4(5)=-25+20=-5$

OK, Joan is getting closer.

Then Joan asks herself, why am I guessing the solution to a math problem when I can ask my math teacher how to solve it?

The point is that there are hard ways to do things, and there are easier ways to do things. The hard way to solve a quadratic equation such as $0 =-x^2+4x$ is to guess. An easier way is to factor.

In this lesson, we will learn different techniques for factoring a wide range of polynomials. Keep in mind as you move through this module that it may seem meaningless to learn how to factor, but it is far more efficient than guessing!