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Solving by Factoring - Problem 1
University of Michigan
Runs his own tutoring company
Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!
Solving a polynomial by factoring. So in this case I have a cubic polynomial behind me and trying to factor, trying to figure out what x value makes this equation true. Whenever we’re dealing with polynomials, it’s easiest to always compare things to 0.
So the first thing I want to do is to get everything to one side. So I always like having my highest degree term, be positive so I’m going to bring everything over to the side of the x³. The first two terms stay the same and then minus 9x minus 18 is equal to 0.
So once we have everything equal to 0, we can then try to figure out how to factor this down. It’s really hard to figure out how to solve something when we’re adding and subtracting, it’s a lot easier when we are multiplying.
So I have four terms here, whenever I have four terms the first thing that pops into my head is grouping. So looking at this I want to group two items that have something in common. So I can look at these first two terms and see that these both have a 16x in common, so I can factor that out of these two, leaving me with x plus 2.
In order for grouping to work, we have to have the same x plus 2, left over here as well when we group these two items. In order to make that happen I have to factor out a -9. Negative because when I distribute this back in, this has to end up -9x, this has to end up -18. So I know that I have to take a -9 out as well.
So now in essence I have a two term, I have something with two terms. This is one term, this is another term, we can factor out what they have in common which in this case is the x plus 2. Factoring out the x plus 2, we need the 16x² to make this first term and a -9 to make this second and you can always check through a distribution that this, if you distribute this through and you the other here you have the same thing. I draw my equal to zeros from both of these but should probably carry it down just to be sure. Okay, so now we have 16x² minus 9.
We know how to factor that, that’s a difference of squares. So the x plus 2 stays the same, this becomes 4x minus 3 and 4x plus 3, and this is equal to 0. Now we have three things being multiplied together to make zero. In order for us to multiply anything together to get 0, one of them has to be 0. So what that tells me is that we could have x is equal to -2, making this term 0. Four thirds, making this term 0, or -4 thirds making the last term 0. So those are our three answers.
And just a recap on how we did that, bringing everything over to one side, factoring out, in this case we had to factor by grouping, and then continuing our factoring down and it looks like I actually did this wrong didn’t I? Looking at it again, 4x minus 3 is equal to 0, let’s double-check that.
We know that 4x minus 3 is equal to zero, if we solve this out, 4x is equal to 3, x is equal to 3/4, okay? I’m sorry, I’m not perfect. And so that actually, I did that twice, did it with the -1 as well so we end up with 3/4 and negative 3/4 is the answer. It’s always good to go back, check your work make sure you don’t make any silly mistakes throughout the entire process.
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