The Importance of Factoring
When you’re trying to solve a simple algebraic expression like $ab = 5b$ for the variable $a$, it quickly becomes second-nature to divide both sides of the equation by $b$, yielding $a = 5$. This makes complete sense, and it’s what most people would do right off, without even thinking. I mean, look at both sides of that equation! If there’s a $b$ on both sides, then the other value on each side of the equation should be equal to each other, giving us $a = 5$.
But not so fast.
What if I were to tell you that this wasn’t the only solution? What if there was another solution to your equation that you missed?
To prove this, let’s look at the original equation again. We have $ab = 5b$, so let’s subtract $5b$ from both sides of the equation. Doing so gives us:
ab - 5b = b(a - 5) = 0.
As you can see, what we did during the second equality was factor out the $b$ from both terms, giving us a product of two terms that equals zero. Once we have that, we know how to solve a product of terms giving zero. At least one of those terms in the product must be equal to zero. Looking at this, we see that $a = 5$ is a solution, like we said before. However, there’s a second solution to this equation, which is $b=0$.
So what happened? How did we miss a solution when we first solved the problem?
The issue was that we divided the equation by $b$, but as we just saw, a solution to the equation is $b = 0$. This means that we were potentially dividing by zero! As most readers know, this is a big problem. We can’t divide an equation by zero, and so by doing this, we were in effect saying that $b \neq 0$. This meant that the solution we found was only valid when $b$ was not zero. As a result, we neglected to think about what happened to the equation when $b$ was zero, and so we lost a solution. By factoring the equation instead of dividing by zero, we can avoid losing the $b=0$ solution and get both in one go.
When we are working through a problem that involves algebra, we tend to push forward without necessarily thinking about the technicalities of what we are doing. Is there the same variable on both sides of the equation? Great, I can cancel them! It’s almost a reflexive habit, but it’s one we need to try and actively resist. By factoring instead of dividing, you create a product that equals zero, allowing you to be sure that you capture all of the solutions to a problem.
Of course, sometimes it’s fine to divide terms out of an equation. However, you need to make sure that you aren’t potentially dividing by zero.