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Multiplying Polynomials: Special Cases - Problem 3
Here I have a product that we might not have seen before up to this point in your class. This one is different because not only is it x minus 2 to some exponent, but that exponent is 3. This is x minus 2 cubed, or to the third power. What that means is it’s the difference of x minus 2 times itself 3 times. Cubed means times itself times itself times itself that’s what it looks like.
So in order to do this problem, I’m going to do this step first and then whatever my answer is, I’m going to multiply it by x minus 2 at the end. Later on in your Math career, you’ll start seeing some short cuts for this maybe memorizing patterns, you don’t have to memorize patterns, you can always do this out by hand, but if you want to be a superstar and a Mathematician, you might want to look ahead in your text book or ask your teacher what the short cuts are here.
Let me show you the long way. So I’m going to go ahead and do this product first, then multiply by that. If I FOIL, my first will look like this, my outers, here come the inners and then plus 4, those are my lasts. When I go through and combine the middle term, it looks this. That’s just this first piece here. I still need to take that whole trinomial and multiply it by the x minus 2, because this represents x minus 2 squared, and then I need to multiply it by x minus 2 again in order to have x minus 2 to the third power.
Okay, so now I’m going to be doing that double distributing. I’m going to use some different colours to help myself stay organized. The first thing I need to do is take x and multiply it by each of these three terms. So x times x² is x to the third and minus 4x² plus 4x. That was all from distributing this x.
Now I’m going to go through and distribute the -2. Be really careful with the negative sign. -2x², +8x and then minus 8 at the end. To combine like terms, here comes my final answer. I’m going to have x to the third take away 6x², by the way this stays as x², it doesn’t become like x to the 0, or anything like that even though I’m subtracting it. Okay minus 6x² then I have plus 12 Xs take away. So there is my final answer for when I had to go through and take x minus 2 and multiply it by itself 3 times.
Again be really careful with your negative signs, make sure you go through it methodically and you should be getting the right answer on these homework problems.