I’m going to give a disclaimer here and say that this was really hard to write out purely in word form. If you’re a visual learner, this will probably confuse you more than anything. It’s not easy to explain mathematical processes without visual aid, so while I tried my best here, I’m not actually sure how effectively I accomplished that. (I’ll also be breaking my typical rule of writing out numbers in sentences just to make things a little easier to read.
Often, in algebra, we have to both factor and expand equations of increasingly complexity. This post will assume you know how to use the “foil” method when expanding equations such as (x + 4)(x – 2). We won’t be talking about expanding today. Instead we’ll look at factoring binomial equations like x² +2x – 8 back into something like (x + 4)(x – 2).
Let’s start simple with x² + 5x + 6. The goal here is to foil backwards. We have to find two factors of +6 that add up to +5. The factors of 6 include 1&6, and 2&3. (Remember we have to look at the factors in pairs. We can’t take 2 and 6 because we’ll be left with 12, which isn’t what we want). 2 and 3 add up to 5, so there we go: (x + 2)(x + 3) = x² + 5x + 6. This is because when foiling it, you’ll end up with x*x + 2x + 3x + 6, and when we combine like terms, this means we can simplify it down to x² + 5x + 6.
Let’s throw in some subtraction to make things a bit more difficult. How about x² – 3x – 28? Again, we look at all the factors of -28 here. In this case we have a few more: 1&28, 2&14, and 4&7. Now, we know one of these numbers has to be negative, because we’re looking for a -28 here. But we can pick which number we want to be negative, so it’s not too daunting. Our target is -3. This means that when adding these two factors together, we’ll be left with a negative number, so the negative will have to be the larger of the two factors. This leaves us with -7 + 4. After that, we can piece it together and get (x – 7)(x + 4) = x² – 3x – 28.
For a leap of faith, one last challenge: adding numbers to x². Let’s factor 3x² – 7x – 20. In a case like this, we have to look at the factors of both 3x² and 20. Obviously, the only two factors of 3 are 1&3. The factors of 20 are 1&20, 2&10, and 4&5. So how do we get all these numbers to add up to -7? This time around order becomes important. Now we have to multiply the factors by each other to get to -7, its not as simple as adding them up anymore.
So, multiplying 1&3 by a pair of 1&20, 2&10, or 4&5, we have to hit -7. One of the numbers in the factors of 20 will also be negative, and again, we want it to be larger than the positive factor. This time, we can solve by trial and error.
Let’s start with 1&3 * 1&20. 3*20 leaves 60, which is way too high for our target. 3*1 leaves 3, and 20*1 leaves 20. This leaves us with either -17 or +17 when we add them together, so the factors of 1&20 are out.
How about 1&3 * 2&10? 3*10 leaves thirty which is still way too big, so we have to multiply 1*10 and 3*2. This leaves us with 10 and 6, and whichever factor of 20 we make negative will leave us with ±4 (± means positive or negative, if you didn’t know).. Much closer, but we need a seven.
So, 1&3 * 4&5 could work. 1*4 and 3*5 gives us 4 and 15, leaving ±11. 3*4 and 1*5 leaves 12 and 5, which adds up to ±7! In order to get -7 specifically, we needed 12 to be negative. Since 3 can’t be negative in this case, this means that we have a -4.
Now, how do we use that information? Let’s back up to 3x² – 7x – 20. We now know that the factors of -20 are -4 and +5, but is the answer (3x + 5)(x – 4) or is it (x + 5)(3x – 4)? Well, for that we look back to the multiplication. In order go get -7, we needed to multiply 3 and 4, which means they cannot be in the same grouping of parenthesis. This means that the answer is (3x + 5)(x – 4) = 3x² – 7x – 20.
There are a lot of special rules and easier ways to learn these techniques than by reading how to do them, though. As I said last week, here is the link to the site that helped me relearn all of this. Though I’m not a math major or anything, I’m generally pretty good at math and am open to help anyone that needs it. There is also a foiling calculator that can solve the problems for you, but if you genuinely don’t know how to do it, use the calculator to check your work, not give you answers. Getting answers to something you don’t understand is the worst thing you can do because it will tell teachers you’re either ready for the next level or you cheated. That said, the calculator is very useful for checking whether or not you got the right answer, and if you already know how to work out the problems, it saves a lot of time on homework.