Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Multiplying polynomials. Our first example is going to be a monomial by a monomial or one single term times one single term. When we have this, what we're going to do is we're going to multiply the coefficients and then each of the like variable portions. So 6 * 1 is six X ^2 * X ^3. Remember, when the variable bases are the same and we're multiplying, we add the exponents. So 2 + 3 would give me five, y * y to the 4th, 1 + 4 is also five, and Z to the 7th times Z understood to the one. If there's not an exponent, 7 + 1 is 8. So monomial times monomial. Literally, we multiply the constants. When we're multiplying variables, we add the exponents. This next example is a monomial by a trinomial. So what we're going to do is we're going to think about the 6X getting multiplied by each term. So six X * 2 X squared, 6X times 3X and six X * -1. Now that minus could have been in front of the six if we wanted to. In the final answer. It won't matter now, we just do the same rules that we had a moment ago. So 6X times 2X12 X cubed, 6X times 3X18X squared, and six X * -1 negative 6X. This next example is a binomial times a binomial. And what we're going to do is we're going to take one of these binomials and split it apart and multiply it by the entire second binomial. So we're going to have three X * 4 X -5 + 2 * 4 X -5 and now we're going to use the skill that we just talked about back here. So 3X times 4X using the distributive property 12 X squared 3X and -5 negative 15X2 and 4X8 X 2 * -5 negative 10, combining like terms, we get twelve X ^2 -, 7 X -10. The next 1A trinomial times a trinomial is going to use the same skills that we just did. I'm going to start over here a little further. So I'm going to have three X ^2 * 4 X squared minus five X + 1, and then plus 2X times 4 X squared minus five X + 1 + -6 * 4 X squared minus five X + 1. So three X ^2 * 4 X squared, 12X to the fourth three X ^2 * -5 X -15 X cubed, three X ^2 * 1 three X squared. The next 12X times 4 X squared is going to give me 8X cubed. I'm actually going to make a column in this case because I'm going to have lots of terms, so I'm going to line up all the cubes on top of each other. So two X * -5 X I'm going to put the -10 X squared underneath the three X ^2 2 X times one. I'm going to start a new column, the -6 * 4 X squared. I'm going to put it in the four or in the X ^2 column -6 * -5 and then -6 * 1. Once I get my columns, I can then just add straight down South 12X to the fourth minus seven X ^3. Let's see, that's -30 One X ^2 + 32 X -6. We've got some for you to try now. Sometimes when we do a binomial times a binomial. Let's go back and grab this one. Three X + 2 * 4 X -5 We have a shortcut now. It only works for binomials times binomials, and it's called foiling. We take the first terms and multiply them together. So 3X times 4X12 X squared. We take the outer terms of the whole thing. So three X * -5 we take the inner full 2 * 4 X, and then we take the last. And once we combine them, we can see that we do get the same answer that we did before. If you have a really good imagination and you squint really hard, you can kind of see Charlie Brown there. There's his little face and his hair and his ears. So foiling. Sometimes I think of as the Charlie Brown method squaring of a binomial A + B * A + B. If we do the Charlie Brown foil method, our first terms together, A * A is a ^2 outer A * B inner B * A, which because of commutative property we can multiply in any order and then B * B. So we get a ^2 + 2 AB plus b ^2. Now this is a formula, so anytime we see something of the quantity A + b ^2, we can use this formula. A -, b ^2 is actually almost exactly the same thing. But what happens is that the outside and inside terms are going to both be negatives because the negative times the positive is a negative, the last one will stay a positive because negative b * -b is positive b ^2. So once again, another formula that looks something like that. So if we have product to sum next, if we foil this, our first terms are going to give us a squared, our outer is going to be negative AB, our inner is going to be positive AB, and our last is going to be negative b ^2. So when we combine our terms here, we end up with just a ^2 -, b ^2. This is frequently called the difference of two squares. So when we multiply a sum and a difference with them both being binomials, we get the difference of two squares. So for four t -, 1 ^2, if we use the formula that said a -, b quantity squared equals a ^2 -, 2 AB plus b ^2, that first term squared would give me 16 T squared. Multiplying these two together and then having it multiplied by a -2, we'd get a positive. No, I'm sorry, multiplying the terms and not paying attention to the sign and then multiplying them by two SO4T times 1 is 4T and then multiplying it by that -2 would get -8 T and then the last 1 * 1 is +1. We can always foil it or multiply it out, and frequently students prefer this method is to just go ahead and do what we've already learned. So when we do this and we combine our like terms, we can see that we get the sixteen t ^2 -, 8 T plus one here. This is the product of a sum and a difference. So it's the difference of two squares, sixteen t ^2 + 12 T -12 T -9, so sixteen t ^2 -, 9. If we saw that this was the formula, we just take the first one and square it, subtract the second one and square it. So 4T squared is sixteen t ^2, 3 ^2 is 9. So now a couple for you to try. This one gets a little more complex, but I'm going to let you try it anyway. You're going to do the first One X + 5 point a ^2 minus the 2nd, and then combine your like terms. The last example I'm going to do is finding a value. So find F of a + H -, F of a. So F of a + H means that every time I see the unknown X, IE whatever was in the parentheses, I'm going to put in a plus H now. So I'm going to get a plus H quantity squared +5. So when I do this, this is really A + H * A + H +5. Foiling out the first terms, we get a ^2 plus AH plus AH. Again, outer term and inner term are the same, so we get a ^2 + 2 AH plus H ^2 + 5. Now that's just this first portion. F of A is just going to be. Every time I see my unknown, I'm going to stick in A. So F of A is a ^2 + 5. So the question asks for what is F of A + H -? F of A. So pulling this all together, we're going to have a ^2 + 2 AH plus H ^2 + 5, all subtracted a ^2 + 5. The A squareds are going to cancel, so we get 2AH plus H ^2 and then the +5 and -5 are also going to cancel. One last one for you to try. Thank you and have a wonderful day.