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# Multiplying binomials by polynomials challenge

CCSS.Math:

## Video transcript

so we've got the expression 2x plus 4 times 5x minus 9 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 couple 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 and there's there's a couple of ways to think about tackling this I'd like to think about it is applying the distributive property twice so this expression on the left hand side we can rewrite it or one way to think about is we distribute the entire 2x plus 4 onto the 5x minus 9 so this is the same thing as 2x plus 4 times 5x plus 2x plus 4 times negative 9 or we could write it like this it is it is 5x 5x times 2x plus 4 2x plus 4 and then we could either view this as a plus negative 9 or just a minus 9 minus 9 times once again 2x plus 4 2x plus 4 and all we've done is we've distributed this 2x plus 4 onto the 5x minus 9 well now when we write it like this when we look at the 5x times the 2x plus 4 we can distribute the 5x we can distribute the 5x onto the 2x plus 4 so what's 5x times 2x well that's going to be 10x squared 5x times 4 is plus 20x plus 20x and then we have and then we have negative 9 times 2x is going to be negative 18x negative 18x and then you have negative 9 times 4 is negative 36 and now we can simplify this a little bit we have two first degree terms so let's see we have 10x 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 X's and I were to take whay 18 of those x's I'm gonna have 2x left over 20 minus 18 2 X and then of course we still have the minus 36 now all I've been doing so far is 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 a x squared plus BX plus BX minus 36 minus 36 and now that I 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 2x here we have BX right over here and so 2 must be equal to B or B must be equal to 2 and it all worked out that our constant terms are the same on both sides so there you have it a equals 10 B equals 2 now once your practice to this you might be able to say well how can I get a faster way to do this although you it might be a little bit more prone to careless mistakes as you can 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 2x times the 5 X and that's going to be 10 x squared and then you could say all right a is going to be equal to 10 and then you can say how can 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 9 so that would be negative 18x or you could multiply 4 times 5x 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 2x so 2 2 X BX B must be equal to 2 and then you can just check well how am I going to get a constant term well I have to multiply these two constant terms for times negative nine you're gonna get negative 36 so the second way I just did it'll be a little bit faster 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