All right, 5-3 for Math 100 is special products. So we did FOIL in section 5-2 and we were multiplying our polynomials. So let's just review FOIL and what that actually means.
So the F stands for the first terms in the binomials. The O stands for the two outside terms. The I stands for the two inside terms and the L stands for the two last terms. Now, I don't want you guys to freak out if you're like, oh my gosh, I don't understand FOIL because really it's just a big distributing problem. So if I multiply X times this X, those are the first two terms in the binomials.
If I multiply X times six, those are the two that are on the outside. And then if I move over to the five and I multiply five times the X, well, those are the two that are on the inside and I multiply 5 times the six. That would be the last two in the binomials, so you can call it foil or you can just call it distributing.
It doesn't matter. So just a quick review. We multiply X times X and get X squared X times the six and we get 6X X times the five and we get plus 5X.
And then we have the 5 times 6 so we get plus 30. Now a common mistake is that I see students multiply these correctly, but they forget sometimes to put the plus symbols in between and those are super important because sometimes they are not addition, it is subtraction. Alright, in the last step here would be combining the like terms. Sir X squared does not have like terms. It comes down 6X plus 5X is 11X and then 30 doesn't have a like term. So this would be our answer for multiplying x plus 5 times x plus 6. So that's just a quick review of FOIL.
One of our special products that we're going to look at is called, actually, you don't really have to give it a name. Let's say we are multiplying two binomials that look almost identical. One of them is an addition and the other one is a subtraction.
Okay, and so really there's no little special rule here. If I just use that foil that we just went over. Okay, so if I multiply A times A, I get A squared.
If I multiply A times that negative B up there, that would give me negative AB. So this little plus that I wrote down right here doesn't belong there. Okay, because a times negative b would be negative ab. Right, and then if I multiply this b times both of those and see what happens, I would get b times a would give me a positive ab.
Now, it's okay if you call that ab or ba. It doesn't matter. You're multiplying. But since I called the other one ab, then I'm going to call this one the same thing because we're multiplying.
Now I multiply b times b, but one of them is positive and one of them is negative. So. Positive times negative is negative b squared. Now our two terms in the middle have exactly the same variables which makes them like terms.
So I bring down my a squared and look what happens. That negative ab plus ab becomes a zero. So I don't have to write down the number zero.
I bring down my negative b squared and so this would be my answer. Now if we just stick with distributing like we did FOIL over here on the left, this all works out no matter what our problems look like. So let's just jump into some problems that aren't just variables like the one I just did that actually have some numbers in them.
Alright, like 4y plus 3 and 4y minus 3. It's very similar to the one I just looked at. Okay, so we're going to multiply this 4y times both of those using FOIL. Then we'll multiply the 3 times both of those. So 16y squared minus 12y. Careful with our signs.
Then when I'm multiplying the 3, I get 3 times 4y plus 12y. minus 9. The two in the middle are like terms, and notice that one of them is plus and one of them is minus, so that gives me 0, and I bring down my minus 9. So my answer here is 16y squared minus 9. Alright, now another thing that I see students mess up on is if you have a binomial that is squared, a common mistake is for students to go. Oh, that's just a square plus B squared. That is not true because A plus B squared means 8 times A plus B times A plus B. And now we're going to do this distributing with our foil.
to see what that gives me. So a times a is a squared. a times b is plus ab.
Now I'm multiplying the b times a and I get plus ab. Remember the order you write ab in doesn't matter but whichever one I write the first one then I'll do the second one that same order. And then b times b is plus b squared. My like terms in the middle notice they have the same sign so we imagine a one sitting in front right here as our coefficient so i have 1ab plus 1ab gives me 2ab and then i bring down my b squared so this would be my solution Alright, so being real careful and don't just call it a squared plus b squared.
Obviously, that would not be correct. And the same thing goes for if that was a subtraction. Don't just say a minus b. Actually, a squared minus b squared.
Let's actually do the FOIL. So let's do a problem with a subtraction that actually has some numbers in it. Alright, let's try I'm just x minus 4 squared. Just x minus 4 squared.
Alright, so x minus 4 squared means x minus 4 times x minus 4. Don't get lazy. Write it down. Multiply it by itself.
Now I'm going to multiply this x times both of those and then the negative 4 times both of those. So x times x is x squared. x times negative 4x is negative 4x.
Okay, now I move to the negative 4. Negative 4 times x, negative 4x, and then negative 4 times negative 4 is positive 16. My like terms in the middle, notice they have the same sign, so I add them, and I get negative 8x and bring down this plus 16. So a common mistake is students would look at this and say, oh, the answer is x squared minus 16, and that is not true because then we're missing that 8x in the middle. So actually, this is all of section 3. It is a fairly short section that really is just more multiplying, more doing FOIL. Now, my math lab does it.
give you these little rules and they tell you that you can memorize the rules but as long as you know foil and you can distribute you can figure out the answers to these and that is more important than just straight memorizing some things so we'll look at section four on our next lesson