Subtraction doesn't make a whole lot of sense if you don't know how to add. And division is pretty tricky if you don't know how to multiply. Likewise, factoring doesn't make much sense if you don't know how to distribute.
Before you can factor, you need to make sure you understand how to simplify a problem like (x + 2)(x + 5). A lot of teachers use the FOIL method to do this (Firsts, Outsides, Insides, Lasts). You can call it FOIL if you want, really you're just distributing twice. First, distribute the x (multiply the Firsts, multiply the Outsides). Then, distribute the 2 (multiply the Insides, multiply the Lasts). Last, combine like terms to get the simplified answer.
Factoring: Going Backwards
You use the distributive property (FOIL) when you have to multiply two binomials like (x+2) and (x+5) above. The answer we got was a quadratic since it has an x-squared term. Factoring is going the other way around. With factoring, you start with the quadratic and go backwards to try and figure out what must be in the two parentheses.
Before we try one, think about where each part of the quadratic came from when we used FOIL to simplify (x+2)(x+5). The x-squared term came from the Firsts - from multiplying the two x's. The 10 came from the Lasts - from multiplying the 2 and the 5. The 7x in the middle came from combining the terms we got from the Outsides and Insides.
In order to factor this, we need to figure out what the two binomials are that multiply to the quadratic we've been given. We need to know ( ? ) times ( ? ) will equal this quadratic expression. We know that the x-squared term will come from multiplying the Firsts - this means that there must be an x in both Firsts spots. We also have all plus signs in the quadratic, so the two binomials will also have plus signs. This gives us:
Now we just need to figure out what goes in the Lasts spots. We know that we have to end up with a 12 at the end of the quadratic for this to work. This means that the Lasts have to multiply to 12. Thankfully, there's only a few combinations of numbers that multiply to 12.
Options for Lasts spots: 1 and 12 2 and 6 3 and 4
Now we can just pick a combination and try it to see if it works. Let's try the 1 and 12 first. Distribute twice (FOIL) to see if it comes out to the quadratic we've been given.
So this combination didn't work. The quadratic we were given has an 8x in the middle and this combination gave us 13x. Since this one didn't work, cross it off and try the next option: 2 and 6.
We found the two binomials that correctly multiply to the quadratic we were given. This means the final answer is (x+2)(x+6). The good thing about factoring is that you can always check your answer. You can use what you know about the distributive property (FOIL) and narrow it down to only a few different possible options. Pick an option and use FOIL to see if it works. If it doesn't, cross it off and move on to the next one. The more you do these, the easier they will get.
The goal is to figure out ( ? ) times ( ? ) equals the quadratic we've been given. We know if we distribute (FOIL), the first term has to come out to the x-squared. This means we'll need an x in both the Firsts spots. This gives us:
The constant at the end of our quadratic is a -15. This means the two numbers in the Lasts spots must multiply to -15. How do you get two numbers to multiply to a negative? One of them must be positive and the other one must be negative. A common mistake that students make on this type of problem is that they see two negatives in the quadratic and assume that the factored answer will have two negatives. If you multiply a negative by a negative, you'll get a positive at the end. Anytime the last number in the quadratic is negative, you'll need one plus sign and one minus sign. This gives us:
Now we need to think about the possible numbers for the Lasts spots. We know they need to multiply to -15 and that one will be positive and the other will be negative. When you write out the options, don't worry about the signs. We can fix that at the end if we need to.
Options for Lasts spots: 1 and 15 (with either the 1 or the 15 negative) 3 and 5 (with either the 3 or the 5 negative)
How do we know which option it is? Just pick one and try it. Let's try the 1 and the 15 first. Don't stress out about which one to make negative. Just try it and see if it works.
So this option didn't work. Let's see if it works if we switch them and make the 1 negative this time.
Any time you just switch the two signs, the only difference in the answer will be that the middle term will change signs. Knowing this fact can save you a lot of time when you're trying to find the factors. Let's say you wanted the middle term to be 10x and you got a -10x. To fix the problem, all you do is switch the signs.
Now that we've ruled out the 1 and the 15 as an option, let's try the 3 and the 5. Remember, don't worry about which one to make negative. If we get a positive 2x in the middle instead of a -2x, all we need to do to fix it is to switch the + and - signs.
Once we switch it to a positive 3 and a negative 5, it works!
This means the final answer is (x+5)(x-3). The more you practice factoring, the easier it will get. You may even start to be able to eliminate certain possibilities in your head if you can recognize ahead of time that the middle number is going to end up too large or too small.
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