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# Factoring using polynomial division

CCSS.Math:

## Video transcript

we are told the polynomial P of X is equal to four X to the third plus 19 x squared plus 19 X minus six has a known factor of X plus 2 rewrite P of X as a product of linear factors so pause this video and see if you can have a go at that all right now let's work through it together so if they didn't give us this second piece of information that has a known factor of X plus 2 this this polynomial would not be so easy to factor but because we know we have a known factor of X plus 2 I could divide that into our expression and figure out what I have left over and then see if I can factor from there so let's do that let's divide X plus 2 into our polynomial so it's 4 X to the third power plus 19 x squared plus 19 X minus 6 and we've done this multiple times already we look at the highest degree terms X goes into 4x to the 3rd 4x squared times I put that in the x squared or the second degree column 4x squared times X is 4x to the 3rd 4x squared times 2 is 8 X 8x squared and then I want to subtract these from what I have up here so I'll subtract and then I'm going to be left with 19 x squared minus 8x squared is 11x squared and then I will bring down bring down the 19 X so plus 19 X and so once again look at x + xi x squared x goes into 11 x squared 11 x times so that's + 11 X 11 x times X is 11x squared 11 x times 2 is 22 X need to subtract these from what we have in that teal color and we are left with 19-22 if something is negative 3 of that something in this case it's negative 3x and then we bring down that negative 6 and then we look at once again at the X and the negative 3x X goes into negative 3 X negative three times and so negative three times X is negative three X negative three times two is negative six and then if we want to subtract what we have in red from what we have in magenta so I could just multiply them both by negative and so everything just cancels out and we have no remainder and so we can rewrite P of X now we can rewrite P of X as being equal to X plus two times all of this business 4x squared plus 11x minus three now we're not done yet because we haven't expressed it as a product of linear factors this one over here is linear but 4x squared plus 11x minus three that's still quadratic so we have to factor that further and let's see there's a couple of ways we could approach it we could use well we could try with something like the quadratic formula or we could factor by grouping and to factor by grouping and the whole reason we have to factor by grouping is we have a leading coefficient here that is not one and so we need to think of two numbers whose product is equal to 4 times negative 3 so we have to think of two numbers let's just call them a and B a times B needs to be equal to 4 times negative 3 which is negative 12 and a plus B needs to be equal to 11 and so the best that I can think of they have to be opposite signs because their product is a negative so if I had negative if I had if I had positive 12 and negative 1 that works if a is equal to negative 1 and B is equal to positive 12 that works and so what I want to do is I want to take this first degree term right over here 11x and I want to split it into a 12x and a negative 1x so let's do that so I can let's just focus on this part on this part right now and then I'll put it all back together at the end so I can rewrite all of this business as 4x squared and instead of writing the 11x there I'm going to use this blue color I'm going to break it up as a 12 X so plus 12x and then minus 1x No these two still add up to 11 X and then I have my minus three and then let's see out of these first two what can i factor out let's see I can factor out a 4x so I can rewrite these first two as and if this is unfamiliar to you I encourage you to review factoring by grouping on Khan Academy so if we factor out a 4x that's going to be it's going to be we're going to be left with an X here and then we're going to be left with a 3 over here and then these second two terms if we factor out a negative 1 so I'll write negative 1 times we're going to have an X plus 3 and so then we can factor out the X plus 3 so let's do that I'm running out of colors so i factor out the X plus 3 and I am left with X plus 3 times x 4x I'm going to keep these colors the same so you know where I got them from 4x minus 1 this is a very colorful solution that we have over here and there you have it I factored the second part into these two factors and so now I can put it all together I can rewrite P of X as a product of linear factors P of X is equal to X plus 2 times X plus 3 times 4x minus 1 for X minus 1 and we are [Music]