Common Core Algebra I.Unit 1.Lesson 8. More Complex Equivalency

Equivalency, before we even jump into exercise one, remember that we think that or we know that two expressions are equivalent. If they always have the same value, no matter what you put in for the value of X or Y or Z or anything else. They are the same, they just look differently. So in this lesson, we're going to explore equivalency that's a little bit more complex than what we've seen in the past. And we're going to start with exercise one. All right, it says consider the product X minus two times X plus 5. It is equivalent to one of the expressions below determine which by substituting in two values of X to check. All right? So see if you can remember how to do this. Figure out is X squared minus ten the same as X minus two times X plus 5 or is X squared plus three X minus ten the same. 

All right? Pause the video now and play around with this a little bit. All right, let's do it. Now this is going to take a little bit of arithmetic, but shouldn't be too bad. We're going to put three in for X and we're going to see what this product is. All right? Three minus two is one. Three plus 5 is 8. So it ends up being 8. Let's put three in here. Now remember order of operations insists that we do this squaring before we do the subtraction. So we'll do three times three and we'll get 9, we'll get 9 minus ten, which is negative one. Uh oh. So these things are not the same. Which means there's no way these two things are equivalent. There's no way. These two expressions aren't equivalent because one we put three in, they don't give us the same value. Now the question is, how in the world does this last expression the same? I mean, I've implied that one of them is equivalent to that first one. 

So let's give it a shot. Let's take three and square it. And then do three times three, and then minus ten. All right, let's do three times three, which is 9. And let's do another three times three, which is 9 minus ten. And then we'll get 18 minus ten. And that's 8. Ah, cool. Green light red light, maybe? Yes. Right? Let's try it with another value of X now we already know the second expression isn't equivalent, but I'm going to test it anyway. All right, so let's do 5 minus two. Times 5 plus 5. 5 minus two is three. Oops, that doesn't quite look like a three, 5 plus 5 is ten. And three times ten. Is 30. Let's do it over here. We'll get 5 squared minus ten, 5 times 5, 25, ten, 15, and again, take a look at these, not the same. 

On the other hand, let's put 5 into this expression, get 5 squared plus three times 5 minus ten, 5 squared is 5 times 5, 25, three times 5 is 15 minus ten. 25 and 15, that's 40. Minus ten gives me 30. Whoops. Oh, I was boxing those, wasn't I? Yes, again. Yes. Right? That's really cool. So somehow it appears, it appears that the product of these two binomials somehow is the same as what's known as this trinomial. Okay? But why? We're going to play around with that a lot today. Okay? So I'm going to clear out the text. Write down whatever you need to. All right, here it goes. Let's keep going. So slightly different multiplication problem. But when we find a product of two binomials like this, okay? We want to understand why it works the way it does. So let's kind of go through this. 

This is probably something you've seen before, but maybe something you haven't justified before. So let's take a look. I take X plus three and I multiply it by X plus 5, and the first thing I did is I took X plus three and I multiplied it by X and then I took X plus three and I multiplied it by 5. What property is that? Can you think of it? Ah, right. It's the distributive property. Distributive property, right? Because I took this one thing, X plus three, and I multiplied it by both quantities on the inside. Now let's see what we did next in step two. Here I took X plus three times X and I multiplied this X by this X that's here. And then I multiplied this three by this X that's here. Then I took X plus three, and I multiplied it by 5. And I did X times 5. And then I did three times 5. Well, that's also the distributive property. 

Now, the last thing that I did was I took these two things. This gets a little bit more confusing. So I'm going to erase a little bit of this. Right? The last thing I did was I took these two things. And I rewrote them as X times three plus 5. Now normally I realize we've been taking this and just saying three X plus 5 X is 8 X I get it. But really when we do that, what we're again using is the distributive property. It may seem like we're doing the opposite of the distributive property. But we're not. The distributive property says a times B plus C is a times B plus a times C but we could have easily written the distributive property this way. A times B plus a times C is equal to a times B plus C we could have written it that way and that's exactly what we're doing here. So a lot of math teachers will call this process, this process of multiplying a binomial by a binomial, a call it. Double distribution. In my mind, it's actually triple distribution. 

But they'll call it double distribution because they distribute the binomial through the first binomial. And then they distribute again. You can almost argue it's like quadruple distribution because in the second distribution step you do it twice. So anyway, whatever, double distribution, sometimes known as foiling, we'll look at both processes as the year goes on. Okay? So let's clear out the text. There it goes. And it's gone. Distribute distributive property is an amazingly important. All right? Write out each of the following as a equivalent trinomials and expression involving three terms. Let's do the first one together. Or maybe even the first couple together. Make sure that we understand the process as well, I'm going to discuss that terminology a trinomial, et cetera. 

So here we go. I'm going to take my X plus 6, and I'm going to multiply it by this X, and then I'm going to take X plus 6, and I'm going to multiply it by three. All right? So I'm distributing the X plus 6 to the X and to the three. Now what I'm going to do is I'm going to distribute this X through here. Now X times X, that's X squared. X times 6, that's 6 X. Three times X, that's three X and three times 6, that's 18. All right? What very often happens then is that these two inner terms are like terms. 6 X plus three X gives me 9 X 18. Now this is what's known as a tri no meal. Try meaning three, no meal, terms. We have one, two, three terms, terms are separated by addition and subtraction. Okay, addition and subtraction, because they come last. So literally the way I look at X squared plus 9 X plus 18 is I've got a term of X squared. A term of 9 X and a term of 18, and I add them all together. 

I can't do anything with them now. All right, but I have a trinomial. Let's do another one. Okay, let's take the X minus four. And distribute it. X minus four times X plus X minus four times 6. All right? Then I can distribute this X X times X is X squared, X times negative four is negative four X 6 times X is positive 6 X 6 times negative four. Is negative 24. These two terms are like terms, negative four, positive 6, leaves me with a net positive two X and then -24. All right, equivalent trinomial. X minus three times X minus three. Let's do it. We're going to do the first X minus three times X and I'm going to include that subtraction here. And then I'll have X minus three. Times three. So here, we'll have X times X, which is X squared, and X times negative three, which is negative three X here, it gets a little bit tricky because I have both subtraction and multiplication. 

So what I'm going to do is I'm going to do three times X, which is going to be three X and then I'm going to apply a negative to it. Here I'll have three times negative three, which is negative 9, but then I have another negative, so that's positive 9. Well, maybe look at another way to do this in a second, but that gives me X squared -6 X. Remember negative three X negative three X positive a negative 6 X definitely not a positive. You only get positives when you're multiplying or dividing. But when you add two negatives, you definitely got a negative. So X squared -6 X plus 9. All right? Let's go down to D now, it's interesting because some people work these problems the way that I've been doing it. Some people will actually start by saying, how about this? Let's first take this two X and let's distribute it to three X plus one. And then let's take this three and distribute it three X plus one. 

Now really, when we do it this way, when we kind of chunk this first piece, what we're really doing is we're actually distributing the three X plus one first, right? Then we can distribute this through two X times three X is two times three, which is 6 X times X, which is X squared. Two X times positive one is positive two X three times three X is three times three, which is 9, and then the X comes along for the ride. And then three times one is positive three. Again, here comes my light terms. Two X and 9 X is 11 X plus three. All right. I hope those two approaches don't confuse you. I'm going to probably do both of them here and there. I'm going to stick with this one. Let me take the three X and multiply it by three X plus two. Now watch what I'm going to do. I'm going to take this, and I'm going to treat it. Like I'm multiplying by negative four. 

It's probably an easier way to think about it than this way. Again, distribute three times three is 9. X times X is X squared. Three X times. Two is three times two, which is 6. The X just kind of comes along for the ride. Negative four times three X is negative. 12 X and negative four times two is negative 8. 9 X squared. Here we've got 6 X -12 X that gives me negative 6 X and negative 8. All right. Extremely important process. If you haven't tried one on your own yet, why don't you try letter F? All right, let's go through it. Again, I'm going to kind of stick with the approach of taking this and multiplying by X -7. And then negative one and multiplying it by X -7. Distribute four X times X would be four times X squared. And four X times negative 7 will be four times negative 7, which is negative 28 and the X comes along from the ride. 

Negative one times X is negative one X or just negative X and negative one times negative 7 is positive 7. So what do I get? I get four X squared negative 28 and negative one is negative 29 X. Plus 7. All right. That's it. That is it. Okay, there's a lot on that page. Pause the video now right now what you need to think about what we did. Here it goes. All right. So let's see if we can attack a pattern by using what we've learned today. Exercise force says Jeremy has noticed a pattern that he thinks is always true. If he picks any number, any number and finds the product of one number larger. And one number smaller than it, the result is always one less than the square of his number. Okay, test some numbers to see if the pattern holds. Okay, so let's I'm going to use N for number. Let's try 5. 

Okay, what did Jeremy do? He took one number larger than 5. And one number is smaller than 5. And he found their pride and he got 24. And he thinks that that's one less than the square of his numbers. So if I square the number and subtract one, we get 5 squared, which is 25. Minus one is 24. Huh. All right. That worked. Let's try, let's try N equals. Let's go with 11 11s kind of annoying number. So what would we do? We do one number larger than it, 12 times one number smaller than it, ten, and we get the product of one 20. Now let's see 11 squared minus one, right? The square of his number minus one, one less than. 11 squared is 121 minus one, which is one 20. Or so why don't you try one for yourself, okay? Pause the video now, pick any number, make it nice, make it nice, I think, and see if this works. 

All right, I'm going to go nuts. I'm going to actually use a negative number. I'm going to go with negative 8. As a little bit harder, because one number more than negative 8 is negative 7 and one number. Less than negative 8 is negative 9. Don't forget a negative times a negative is a positive, and we get 63. Now we have to be very careful. Negative 8 squared minus one, negative 8 times negative 8 is 64. Subtract one, and we get 63. Because these are the same. So so far it looks like Jeremy's pattern is holding. Letter B says, give an algebraic explanation that shows that jurors may pattern will work for any number. Use let statements clearly define your variables. All right, watch this. So let N be equal to any number. Okay. Let N plus one be equal to one more. Then N and we'll let N minus one be equal to one less than all right, isn't that the case? You know, to get one more than N, we'll do N plus one to get one less than N we'll do N minus one. Now we want to find the product of those two. Isn't this exactly what we did above? So we want to find their product. All right, let's do what we did in the previous exercise. 

We'll take this end and we'll multiply it by N minus one. Plus one times N minus one. Distribute N times N is N squared N times negative one is negative N one times N is positive N and one times negative one is negative one. Now it's interesting because a negative N and a positive N they just cancel, right? And we get N squared minus one. But that is exactly what we wanted, right? This is one less. Than the square of the number. And that's kind of cool. One less than the square of the number. So we've just proven that Jeremy's pattern doesn't just hold for three numbers, but it holds for all numbers if I take any number I want, even something ugly, like pi or some terrible looking square root, and I add one more than it, and one less than it, and I multiply them together. And then we'll get the same thing as if I just squared the number and subtract it one. Kind of cool. All right, I am clear in this app. 

All right. So exercise 5 little multiple choice question. Which of the following expressions is equivalent to the product X minus two times X minus four, show the calculations that you used to find your choice and test using a value of X all right, see what you can do. Hit pause on the video and play around with this. All right. Well, I hope you didn't forget to do the test as well, because that really ensures you got it right. I'm going to rewrite the problem down right here. X minus two times X minus four. I'm going to take this X and multiply it by X minus four, and then I'm going to take this negative two and multiply it by X minus four. All right? Distribute, and I'll get X times X, which is X squared, and X one too many arrows there. That'll redundancy there. X times negative four is negative four X. Negative two times X is negative two X and negative two times negative four. Is it positive 8? Negative four X and negative two X combined to be negative 6 X and of course positive 8 is positive 8. So it appears to be choice three. But let's also do the quick test. All right, remember doing these in previous lessons. 

Let's do a test. I don't know. Let's test X equals two. Let's not test X equals two. Well, actually, let's test X equals two. It doesn't matter which one we pick. Let's test first X minus two. Times X minus four. All right? If we put two in there, we'll get two minus two times two minus four, two minus two is zero. Two minus four is negative two. And any time we multiply by zero, we get a result of zero. That's kind of why I was back and out of using two, but I like it anyway. Now let's test X squared -6 X plus 8. Let's test this one. So we're going to put two in and square it. Put two in and multiply by negative 6 or by positive 6 depending on how you look at it. Two times two is four, 6 times two is 12. Plus 8. Four -12 is negative 8. Plus 8, negative 8 plus 8 is zero. Yes. All right, they are equivalent. Good little check there. Okay? So pause the video now if you need to. All right, here we go. Let's finish up. 

So today we looked at more complex equivalency and one thing that you really need to come out of today with is at least an initial ability to multiply two binomials together, right? This double distribution process. Your teacher may have taught you how to foil. That's completely okay. It's just a way of making sure that everything multiplies everything else. And that you combine correct terms. But more importantly, we also explored the idea of equivalency more. The idea that two expressions can look completely different and yet give you exactly the same value no matter what value of X you put in. Okay? So thank you for joining me for another common core algebra one lesson by E math instruction. My name is Kurt weiser. And until next time, keep thinking and keep solving problems.