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Multiplying binomials by polynomials challenge

Sal finds the values of coefficients a and b that make (2x+4)(5x-9)=ax²+bx-46 true for all x-values.

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Video transcript

- [Voiceover] So we've got the expression two X plus four times five X minus nine is equal to AX squared plus BX minus 36. And what we want to figure out is what are A and B going to be? And I encourage you to pause the video and try to figure it out. Well there's a coupe of ways of trying to tackle it and the most straightforward would be just let's multiply these two binomials on the left-hand side and let's see if we can match up the terms and match up the coefficients. So let's multiply this left-hand side. There's a couple of ways to think about tackling this. I like to think about it as applying the distributive property twice. So this expression on the left-hand side we can rewrite it, or one way to think about it is we distribute the entire two X plus four onto the five X minus nine. So this is the same thing as two X plus four times five X, plus two X plus four times negative nine. Or we could write it like this. It is five X five X times two X plus four. Two X plus four. And then we could either view this as a plus negative nine or just a minus nine. Minus nine times, once again, two X plus four. Two X plus four. And all we've done is we've distributed this two X plus four onto the five X minus nine. Well now when we write it like this, when we look at the five X times the two X plus four, we can distribute the five X. We can distribute the five X onto the two X plus four. So what's five X times two X? Well that's going to be 10 X squared, five X times four is plus 20 X. Plus 20 X. And then we have, and then we have negative nine times two X is going to be negative 18 X. Negative 18 X. And then you have negative nine times four is negative 36. And now we can simplify this a little bit. We have two first degree terms. So let's see, we have 10 X squared, 10 X squared, and then these two first degree terms, let me circle them. So we have these two first degree terms. If I have 20 Xs and i were to take away 18 of those Xs I'm going to have two X left over. 20 minus 18, two X. And then of course we still have the minus 36. Now all I've been doing so far is simplifying or rewriting the left-hand side. We have to remember, this was an equation so this needs to be equal to the right-hand side. So this is going to be equal to AX squared. So AX squared plus BX, plus BX minus 36. Minus 36. And now that I've wrote it a little color-coded it might jump out at you what A and B are going to be. We have 10 X squared over here and then the second degree term on the right-hand side is AX squared. So 10 must be equal to A, or these two coefficients must be equal. So we could write A is equal to 10. And then when we look at the first degree term, we have two X here and we have BX right over here. And so two must be equal to B, or B must be equal to two. And it all worked out that our constant terms are the same on both sides. So there we have it, A equals 10, B equals two. Now once you're practiced at this you might be able to say well how can I get a faster way to do this? Although it might be a little bit more prone to careless mistakes. Is you could say well how can I get, how do I get an X squared? How do I, when I multiply these things out, how do I get an X squared? Well the only way that I can get an X squared is when I multiply the two X times the five X. And that's going to be 10 X squared. And then you could say, alright, A is going to be equal to 10. And then you could say how could I get an X? Well there's two ways that you can get, two ways to get an X. You could multiply two X times negative nine. So that would be negative 18 X. Or you could multiply four times five X, which is going to be plus 20 X. If you add these two together you're going to get, they're going to be equal to two X. So two. Two X, BX, B must be equal to two. And then you can just check, well how am I gonna get a constant term? Well I have to multiply these two constant terms. Four times negative nine, you're gonna get negative 36. So the second way I just did it will be a little bit faster, you're a little bit more prone to making careless mistakes, but hopefully you appreciate that, I'm really just doing the same thing. Maybe with different levels of clarity.