Algebra

Polynomial division | Polynomial and rational functions | Algebra II | Khan Academy

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. In this presentation, we will learn the division of polynomials At times, this is called a long division But you’ll see what I’m talking about When we solve some examples Let’s say I want to divide 2x + 4 On the 2nd We do not change the value Rather, we change how the value is explained So we already know how to simplify it We have done this in the past, we can divide the numerator The denominator is 2, and therefore How much is this? This is equal to x + 2 – let me write it this way – Equals – if you divide this by 2– Equal to x Divide 4 by 2, get 2 If you divide 2 by 2, you get 1 So this is equal to x + 2, and this Very direct, I think Another way is that you can extract coefficient 2 here And then these are deleted But I will also show you how to solve it using long division algebra It is an exaggeration on this issue But I want to show you that he does not give us Something new It is not only a new method, it is also useful In more complex matters So you can also write this as (2x + 4) ÷ 2 How much is outside division? You can do this in the same way that you would use it In traditional long division You will say 2 – It always starts with the phrase “top class” – The highest degree phrase ÷ 2 You will ignore the 4 How much is 2x ÷ 4? Well, 2 ÷ 2x = x and put The x is in the x position x x 2 = 2x It will now be floated, as in the traditional long division Will be released now So how much is 2x + 4 – 2x? It’s 4, right? Then how much is the product of 4؟ 2? Equals 2, positive 2 We put it in the position of constants 2 x 2 = 4 We subtract, and the rest is 0 Perhaps this seems exaggerated to For this issue you already know how to solve it You solve it in several steps We will now see these Generalizable process You can do this with a polynomial of any degree divided On the polynomial of another degree Let me show you what I’m talking about Let’s assume that we want to divide x + 1 by x ^ 2 + 3x + 6 What do we do here? So we look at the term higher score here, which is x, and you see the phrase higher order here It is x ^ 2 You can ignore everything else This actually simplifies the process It says how much by dividing x ^ 2 by x? Well, x ^ 2 = x = x, right? x ^ 2 ÷ x = x You put it in the x position This is the x position here or Status x ^ 1 So how many x x (x + 1)? x x x = x ^ 2 x x 1 = x, so x ^ 2 + x And now we pose, as we did here What do we get? x ^ 2 + 3x + 6 – x ^ 2 – Let me be Very careful – this equals -x ^ 2 + x I want to make sure that the signal is negative – It is applied to all of this So x ^ 2 – x ^ 2, this is omitted 3x becomes -x Let me put this signal here So this equals -x ^ 2 – x, so that we can be clear We put all this out 3x – x = 2x And then we go down to 6, or 6 – 0 nothing So 2x + 6 Now, you’re looking at the top notch phrase, x and 2x How much is 2x divided by x? Equal to 2 2 x x = 2x 2 x 1 = 2 We get 2 x (x + 1) = 2x + 2 But we’ll subtract this from this We will put them Instead of writing 2x + 2, we can write -2x + 2 and then we add them together These two are deleted 6 – 2 = 4 How much is the output of 4 x? We can say this is 0, or you can say 4 The rest is 4 is the rest If we want to rewrite x ^ 2 + 3x + 6 / x + 1 – Note this is equivalent to x ^ 2 + 3x + 6 + x + 1, this is divided by this Now we can say this is x + 2 It’s equal to x + 2 + the rest divided by x + 1 + 4 / x + 1 This and that are equivalent And if you want to be sure of that, if you want to move from So to this, what you can do is multiply this by x + 1 / x + 1 and the two are combined So this is x + 2 And I’ll multiply that by x + 1 / x + 1 This is equivalent to multiplying it by 1 Then we add 4 / x + 1 to this I did this to get the same standardized denominator And when you do this addition, when Binary times these two borders, then add the 4 here You should get x ^ 2 + 3x + 6 Let us solve another issue It is fun Let’s assume that we have – we want to simplify x ^ 2 + 5x + 4 / x + 4 So again, we can do a long, forced division We can divide x ^ 2 + 5x + 4 by x + 4 Once again, we follow the same process Look at the phrase highest score in both of them How much is the division of x ^ 2 by x? Equal to x We put it in the x-place This is the x position x x x = x ^ 2 x x 4 = 4x And then of course, we will ask This one from here Let’s put a negative sign here And then we delete this one 5x – 4x = x 4 – 0 = 4 x + 4, then you can see this output You could say x + 4 + x + 4 = 1 Or if you do not look at the constants You will say, well How much is outside the division of x by x? Well, it’s 1 + 1 1 x x = x 1 x 4 = 4 We will throw them from the top, so they will be deleted We have no rest So this simplifies to – this Equals x + 1 There are other methods that you can follow You can try analyzing the numerator into its factors x ^ 2 + 5x + 4 / x + 4 This rewards what? We can analyze this numerator to (x + 4). X (x + 1) 4 x 1 = 4 4 + 1 = 5, all divided by x + 4 These are omitted and we have x + 1 left Any method we follow will succeed, but long division It will always succeed, even if you cannot delete it Factors like this, even if you have the rest In this case, we have no rest This was equal to x + 1 Let us solve another issue to make sure you are Because this is very useful Get it in the gadgets Let’s assume I have x ^ 2 – let me Change it– Let’s suppose I have 2x ^ 2 – I can make this Setting up quickly – 2x ^ 2 – 20x + 12 ÷ – Actually, let’s Make it interesting, until I explain it to you Always succeed I want to move to a superscript statement So let’s suppose we have 3x ^ 3 – 2x ^ 2 + 7x – 4, and we want to divide it by x ^ 2 + 1 I did this on top But we can do a long division process for you We find the output or what Simplify this x ^ 2 + 1 divided by all of these above, i.e. 3x ^ 3 – 2x ^ 2 + 7x – 4 Again, we look at the phrase “higher grade” How much is outside the division of 3x ^ 3 by x ^ 2? Well, it’s 3x We multiply 3x by this, so we get 3x ^ 3 Divide it by 3x So we have to write the 3x here in the x position. if Outside the division is 3x, like this Now let’s hit 3x x x ^ 2 = 3x ^ 3, right? (3x x x ^ 2) + (3x x 1) So here we have 3x I’m sure to put it in the x-place We will put them What do we get? What do we get when we do this? These are deleted We have -2x ^ 2 Then 7x – – because I subtracted 0 from here– 7x – 3x = positive 4x, and we have -4 Once again, look at the phrase top class x ^ 2 and -2x ^ 2 -2x ^ 2 ^ x ^ 2 = -2 -2, we put it in the position of constants -2 x x ^ 2 = -2x ^ 2 -2 x 1 = -2 -2 Now, we’re going to put this from there, so Let’s multiply them by -1, or This becomes positive These two are deleted 4x – 0 = – Let me switch colors – 4x – 0 = 4x -4 – -2 or -4 + 2 = -2 Then x ^ 2, now this has a score greater than 4x, and The top notch is here, so we consider it as the rest We consider it as the rest So this phrase we can rewrite to be equal to 3x – 2 – that’s the 3x – 2– + the rest So 4x – 2, all divided by x ^ 2 + 1 x ^ 2 + 1 I hope you found this offer interesting

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