Algebra

Dividing polynomials 1 | Polynomial and rational functions | Algebra II | Khan Academy

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“Divide (x ^ 2 – 3x + 2) by (x – 2).” We will divide this into this. And we can do that the same way we do initially learned the long division. So how many times (x – 2) is contained in (x ^ 2 – 3x + 2). Another way we could write the same expression is (x ^ 2 – 3x + 2), all this on (x – 2). This, this and this are equivalent expressions. To do this kind of long division – we can call it algebraic long division – you want to look at the member of the highest degree in (x – 2) and the member of the highest degree in (x ^ 2 – 3x + 2). Here is x, and here is x ^ 2. How many times x is contained in x ^ 2? Or how much is x ^ 2 divided by x? This is simply equal to x. x is contained in x ^ 2 x times. And I will write it in this column here above all other articles x. And then we want to multiply x by (x – 2). This gives us – x by x is x ^ 2; x by -2 is -2x. And just as he initially learned in the long division, you want to get that out of this. But this is exactly the same as combining the opposite or multiply each of these terms by -1 and then collect. Let’s multiply this by -1. And -2x by -1 is 2x. Let’s gather now. x ^ 2 and -x ^ 2 are destroyed. -3x + 2x e -x. And then we can download this 2 here. This is a remainder (-x + 2), with the quotient being only x. Then we can say: “Can (x – 2) enter (-x + 2)?” x is contained in -x -1 times. You can see this here. -x, divided by x, is -1. These are shortened. So -1 over (x – 2) – you have -1 by x, which is -x. -1 over -2 is +2. And we want to get that out of this, just like you do in the long division. But it’s the same thing as combining the opposite or multiply each of these terms by -1 and then collect. -x by -1 is + x. +2 over -1 is -2. These are abbreviated and give a sum of 0. These give a sum of 0. We have no residue. We have obtained that this is equal to (x – 1). And we can be sure. If we multiply (x – 1) by (x – 2), we have to get this. So let’s do this. Let us multiply (x – 1) by (x – 2). Let’s multiply -2 by -1. This gives us +2. -2 in x is -2x. Let’s multiply x by -1. This is -x. And then x over x is x ^ 2. Then we gather all such members. x ^ 2; -2x – x is -3x. Then 2 plus nothing is just 2. And again we got this polynomial.

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