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CCSS.Math: ,

we're told the polynomial P of X which is equal to this has a known factor of X plus 6 rewrite P of X as a product of linear factors pause this video and see if you can have a go at that alright now let's work on this together because they give us one of the factors what we can do is say hey what happens if I divide X plus 6 into P of X what do I have left over it looks like I'm still going to have a quadratic and then I'll probably have to factor that somehow to get a product of linear factors so let's get going so if I were to try to figure out what X plus 6 divided into X to the 3rd + 9 x squared and now we're gonna have to be careful you might be tempted to just write minus 108 there but then this gets tricky because you have your third degree column your second degree column you need your first degree column but you just put your zero degree your constant column here so to make sure we have good hygiene we could write plus 0x and I encourage you to actually always do this if you're writing out a polynomial so that you don't skip that place so to speak minus 108 and so then you say all right let's look at the highest degree terms X goes into X to the 3rd x squared times x squared times 6 is 6 x squared x squared times X is X to the 3rd we want to subtract we've done this multiple time so I'm going a little bit faster than normal those cancel out 9x squared minus 6x squared is 3x squared bring down that 0x and then how many times does X go into 3x squared well it goes 3x times and we would write it in this column and notice if we didn't keep this column for our first degree terms we'd be kind of confused where to write that 3x right about now and so 3x plus times 6 I should say is 18x 3x times X is 3x squared we want to subtract what we have in that I guess that color is move light purple not sure and so we get 3 X Squared's cancel out and then 0x minus 18x is of 18x bring down that negative 108 and so then we have X goes into negative 18 X negative 18 times negative 18 times 6 is negative 108 that was working out nicely negative 18 times X is negative 18 X and then we want to subtract what we have in this not-so-pleasant brown color and so I will multiply them both by negative and so I am left with 0 everything just cancels out and so I can rewrite P of X I can rewrite P of X as being equal to X plus 6 times x squared plus 3x minus 18 but I'm not done yet because this is not a linear factor this is still quadratic so let's see can I think of two numbers that add up to three and then when I multiply 2 I get negative 18 so we'll need different signs and then the obvious one is is positive 6 and negative 3 and if that what I just did seems like voodoo to you I encourage you to review factoring polynomials but this I can rewrite because negative 6 plus or actually I said say positive 6 plus negative 3 is equal to 3 and then positive 6 times negative 3 is equal to negative 18 so I can rewrite this as X plus 6 times X plus 6 times X minus 3 and so there we have it we have a product of linear factors and we are done