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Welcome to Algebra I: Products of a Binomial and a Trinomial

24 minutes

Hey, guys, welcome to Algebra I. Today's lesson focuses on finding the product of a binomial with a trinomial. The distributive property helps you get through these problems. You ready to get going? Let's go. Okay, to help you understand how to multiply a binomial times a trinomial, I first need to throw back to multiplying two binomials.

(female describer) On the screen: Find the product. x plus 3 times x plus 7.

(teacher) We've been using FOIL as a method to get through these problems. FOIL was just a shortcut around the distributive property. It is the distributive property in action without drawing all of those loops and hoops. Let's think back. Using FOIL, you multiply the first two terms together. Right? So, x times x, which gives us x squared. Then we multiply the outer terms together, so that was x times 7-- in this case, 7x. Okay? We'd go to the 3, in this case-- multiply 3 times x, those inner terms: 3x. Then we multiply the 3 times the 7, the last terms, and get 21. So FOIL is the same thing as the distributive property. That x squared, that came from multiplying x times x. That was where the "F" came from. Then we multiplied or distributed x to the 7, multiplying those together. Those were our outer terms. The 3x came from multiplying these together, or distributing the 3 to the x. That's where that term came from. Then the last term came from distributing the 3 to the 7. Okay? Essentially, FOIL is the same thing as the distributive property. To finish this problem off, we have some like terms we can combine-- 7x plus 3x gives us 10x. So, x squared plus 10x plus 21. We'd use FOIL to get that answer. The problem arises when you're not multiplying two binomials together-- like here, you're multiplying a binomial times a trinomial.

(female describer) x plus 3 times 2 x squared plus 4x minus 1. In that case, you can't use FOIL because you don't have two binomials, so you use the distributive property. There's not really a shortcut around this one, unfortunately. You have to distribute to get to your answer, okay? I start by drawing the hoops to keep us straight. If you don't need them after a while, stop writing them. I'm going to take the x, and I'm going to distribute it to each term in my trinomial. I'm ignoring this 3 for a second. I pretend like the 3 is just not even there. So, x times 2x squared-- I'll show some work down here. Okay, that's 2x to the 3rd power. Now I have x times 4x. So x times 4x. That's 4x squared, right? So, plus 4x squared. Then x times negative 1. So x times negative 1. So, that's negative 1x, right? So minus x. You've handled distributing the x throughout these terms. Now you have to distribute your 3 to each of these terms as well. That's how we get through this problem. I will get rid of this work so we have a little room to work, okay. Okay. Now I handle the 3. I'm going to distribute that 3 to each term in my trinomial. So I've got 3 times 2x squared. That's 6x squared, okay? I'm going to write that right up here, 6x squared. Now I have 3-- Oop, get that pen back; it escaped us somehow. All right, there we go. Okay, so 3 times 4x, right? 3 times 4x. Well, that's 12x, right? So I've got plus 12x right up here. I need to distribute that 3 to this term, that negative 1. So, 3 times negative 1. That's negative 3, right? So where I'm writing my answers--lost that a second-- I'd have minus 3.

(female describer) She's rewritten the problem as 2x cubed plus 4x squared minus x plus 6x squared plus 12x minus 3. Now you're done distributing everything in your problem. You're at the point where you combine like terms. I get rid of this scratch work so we can just focus on combining like terms. I'll get rid of those hoops as well. Okay. Let's see where we can combine like terms here. We're writing the answer in standard form. We start with the exponent with the largest degree and work our way down. We'll start with the largest degree, the biggest exponent, basically. So my largest exponent in this problem is 3, and I have one term that's raised to the 3rd power. There's nothing to combine that with. That will just stay 2x to the 3rd. I scratch it out to hint I've dealt with it. So now I'll look for exponents of 2, and you scan your terms and see you have two. You have 4x squared, and you have 6x squared. So 4x squared plus 6x squared-- that's 10x squared, right? Okay, so plus 10x squared. I've handled this now. So, now let's look for powers of 1. I see I have a negative x and a positive 12x. So combine those. So, negative x plus 12x, that's 11x, okay? I've dealt with that. I have this constant term on the end, this minus 3. It's my only constant term. There's nothing I can combine it with. So I'll just write minus 3. And after all that work, you're all done.

(female describer) 2x cubed plus 10x squared plus 11x minus 3. The key to that is using the distributive property, keeping everything really organized, and working your way through the problem. Let's try another one. All right, let's bring this up a bit. Okay, so I have x minus 5 times x squared minus 3x plus 2. You remember your best friend, the distributive property. That's how we get through this. I ignore this minus 5 for a minute-- pretend like it's not there-- and just focus my attention on distributing this x. So I'm going to multiply x times x squared. I'll do some work here. So x times x squared-- that's x cubed, right? So x cubed. Then I'll have x times negative 3x, so x times negative 3x. That's negative 3x squared, right? So minus 3x squared. I distribute this x to this positive 2. Can't leave it out. So x times 2-- that's 2x, okay? So plus 2x. And you are all done distributing that first term in that binomial, that x. So now you distribute that negative 5 to each of these terms in this trinomial. I'm going to give us room for scratch work. I show you my work so you know what's going on, how I'm getting these answers. If you can handle that mental math and just write your answers out, feel free. Let's distribute that negative 5 throughout those terms. So, negative 5 times x squared... That's negative 5x squared, right? So up here, I'll have minus 5x squared. I distribute negative 5 to the next term, to this negative 3x. So negative 5 times negative 3x. Negative 5 times negative 3x. So, that is 15x, right? So, plus 15x. And then the last term. Now I need to distribute this negative 5 to this positive 2. So negative 5 times 2-- that's negative 10. So, last term in my row-- I keep moving our screen out of the way--minus 10. Okay, we're at the point where we combine like terms and clean this problem up. Let's get some room to work here. Get scratch work out of the way. You're working on paper; don't erase it. But I want you to have a clear view. That's why I erase my work. Okay, remember, I'm starting with the largest exponent. I want my answer written in standard form. So, I scan my terms-- the largest exponent's a 3. There are no terms besides this first one raised to the 3rd power, so I just carry that along. It's x cubed. I'll cross it out so I know I've handled it. Working my way down as far as exponents, I look for exponents with a value of 2, and I see two. I have negative 3x squared and negative 5x squared. So negative 3x squared minus 5x squared-- that's negative 8x squared. So I'm cool with that one now. I look for exponents of 1-- remember, you don't see exponents of 1-- so I know where they have to be hiding. I have 2x and 15x-- so 2x plus 15x, that's 17x, okay? I'll cross them out because I've handled them. The last remaining term is this constant term, this negative 10, so I know I'll have minus 10 because there's nothing else I can combine it with. You look at your masterpiece. You're all done. All right? Okay, let's try one more together.

(female describer) x plus 1 times x squared minus 4x plus 3. All righty, remember, we're using the distributive property to get through these. So I'm going to ignore that 1 to start with and just focus on x at first. I'm going to distribute this x throughout each of these terms. So, x times x squared-- show my work down here. That's x cubed, okay? I have x to the 3rd power. Now I need to distribute this x to this negative 4x. So x times negative 4x-- That's negative 4x squared, okay? So, minus 4x squared. Now I need to distribute this x to this 3. So, x times 3. x times 3, that's 3x. So, plus 3x, okay? I'm completely done distributing the x. I've worked it through each term. I need to multiply each term by 1 so that I can distribute that 1. Let's get this work out of the way so we have some more room. Get the pen back, all right. We have 1 times x squared; we need to handle that first. So, 1 times x squared-- that's x squared. Okay? Now we have 1 times negative 4x. So, 1 times negative 4x-- that's negative 4x. So, minus 4x. Then we have 1 times 3 to get it to that last term-- 1 times 3; that's just 3. Not surprising that no terms change because multiplying by 1 doesn't change anything, okay? Now we're done distributing. We need to combine like terms, right? Remember, this is where I get us some more room to work. All righty, get the pen back, okay. I look for my largest exponent first. I want my answer in standard form. The largest exponent is 3. There's no other terms that have an exponent of 3, so I'll carry that along. So x cubed. I've handled it; I'll cross it out. I look for exponents of 2. I have this negative 4x squared and this positive x squared. So, negative 4x squared plus x squared-- that's negative 3x squared, so minus 3x squared. You've handled those; let's cross them out. All righty, so now we look for x-- terms that just have x. I see I have 3x; I have this negative 4x. So, 3x minus 4x, that's negative 1x. So I'll write negative x. I'll cross those out because I'm all done. I have a constant term, the only term left. There's nothing to combine it with. So plus 3, and you're all done. You've reached your answer. It's already in standard form for you. Okay? All right, now it's your turn to try. Press pause, take a few minutes. Work through these problems. When you're ready to compare your answers, press play.

(female describer) Find each product. 1. x minus 4 times 2x squared plus x minus 1. 2. x plus 6 times x squared minus 3x plus 2. Let's see what you got. For the first one, x minus 4 times 2x squared plus x minus 1-- You should have 2x cubed minus 7x squared minus 5x plus 4. A lot of terms, but that should be your answer. The number 2, x plus 6 times x squared minus 3x plus 2-- Your answer there should have been x cubed plus 3x squared minus 16x plus 12, okay? If you need to see how I got those, stay with me, and I'll show you. So just like we've been doing, it's all about the distributive property. Start with the first term in your binomial and pretend like second term is just not there. Multiply that first term times each term in that second group. So x times 2x squared... That's 2x cubed. All right, so I know the first term, 2x cubed. Now I have x times x, right? So x times x; that's just x squared. So, plus x squared. Then the last term in this one, x times negative 1. That's just negative 1x. I'll just write negative x, so I'll bring that up here. You're all done distributing that first term. You've got to do the same thing with the second one. I'm going to get us some space up here. All right, let's handle that second one. Multiply negative 4 times each term in the second group. So, negative 4 times 2x squared... That is negative 8x squared. I'll just bring that up here, so minus 8x squared. Now I have negative 4 times x, distributing it to the second term. So negative 4 times x-- that's negative 4x, right? So, just minus 4x. Then the last term, negative 4 times negative 1-- negative 4 times negative 1. That's a positive 4. So, my last term over here is plus 4. We've got to clean it up. We've got to combine like terms; get it all nice. Let's get this-- get the-- Get that out of our way, even those out, so we can just focus on combining like terms. Look for the largest exponent first because you want your answer in standard form. The largest exponent is 3-- the only term that has an exponent of 3-- so 2x cubed, okay? Cross it out because I've handled it. Look for exponents of 2. I see I have x squared and negative 8x squared. If I combine those, I get negative 7x squared, so minus 7x squared. Cross them out because I've handled them. Next I see I have negative x and a negative 4x. When I combine those, I get negative 5x, okay? Cross them out because I've handled them. The last term left is 4-- nothing left to combine it with, so plus 4. All right, and that's how I got that first answer. It's all about the distributive property and then just combining your like terms when you're done. Let me show how I got the second one. I did the same thing, distributive property. So x times x squared, right? So x times x squared-- that's x cubed, okay? So, I know that was my first term. Now I have x times negative 3x, so, x times negative 3x. That's negative 3x squared, right? So, minus 3x squared. Then x times 2 to get to that last term. So x times 2; that's just 2x. So, plus 2x. You've handled that first term, distributing it throughout. So now let's handle the second one. Let's get us some space up here, okay. I'm going to handle the 6. So, 6 times x squared. That's 6x squared, right? So plus 6x squared. Now I need to multiply 6 times that negative 3x to keep working through over there. So 6 times negative 3x; that's negative 18x. So, minus 18x-- squeezing it in over there. And now I need to multiply the 6 times the 2. So 6 times 2, that's 12. Oh, can I make it fit? So plus-- I cannot--plus 12. We can combine like terms. We just need to clean this up. All right, let's get this out of our way. Okay, let's clean this up. You start looking for your largest exponent first, right? I have one term with an exponent of 3, the largest one-- can't combine it with anything. So, just x cubed. And I'll cross it out to note I've handled it. I look for exponents of 2. Negative 3x squared plus 6x squared-- that's a positive 3x squared. All right, done; cross those out. Now 2x and negative 18x, combining those, I'd get negative 16x, so minus 16x. Cross it out because I've handled it. Your last term is that constant number, that 12. Nothing left to combine it with, so plus 12. The magic has happened. You're all done, okay? I hope you're feeling good knowing how to multiply a binomial times a trinomial, and you saw how important the distributive property is to those problems. See you soon. Bye, guys.

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In this program, students will learn how to find a product by multiplying a binomial times a trinomial. Part of the "Welcome to Algebra I" series.

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Runtime: 24 minutes

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