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Write the partial fraction decomposition of the rational expression_ Check your result algebraically: X+ X -...

Question

Write the partial fraction decomposition of the rational expression_ Check your result algebraically: X+ X -

Write the partial fraction decomposition of the rational expression_ Check your result algebraically: X+ X -



Answers

Write the partial fraction decomposition of the rational expression. Check your result algebraically. $$\frac{1}{x^{2}+x}$$

We want to break down this rational function into its partial fraction. First of all, answers are these three questions. Is it improper? Improper means the numerator highest power is greater or equals to the denominator. Has power in this case is not because the highest, powerful, numerous zero, because constant and the highest powerful denominator is too. So it's less than the denominator. So it's a perfection now. In any case, if your partial fraction requires you to do long division, it is in problem. Can the denominator be fact arised in this case? Yes. So let's just spectrum eyes it. Yeah, and are there any repeat of factors? Now there's no it. It looks like this year we pick it would be something like this. We're explains the repeat that one. But in this case, no, there isn't. So now we can break it down into the partial fraction we will write over X plus over X plus one. Now what do we put in the numerator here? Over here, the numerator will be one power less than the denominator since the denominator is power one. So the numerator will be just a constant. So be a same thing over here. Experts. One highest powers. One. So the new morita just be a constant, which is B. So this will be no answer, okay?

In this question. We want to break down this rational fraction into this partial fraction. You, Let's go through the three points. Is destruction improper? Improper will mean the numerator, highest order or highest degree its creator equals to the denominator. Highest degree. Now, in this case, the new Morita has degree zero and Latino Mehta has degrees too. So it is a proper fraction because the numerator power highest power is less than the generate the highest power. Now, in any case, if your fraction is improper, please perform long division first, second point Can the denominator be characterized in this case? Yes, it can affect arrived. So let's do so that point. Are there repeated factors In this case? No repeater will look like this X square. There will be the factor. But in this case, in our questions no, we do not have any with the factors because they will affect the way we decompose it. So now, less Let's this key over X plus over X plus one. Now what do we put the numerator? Emerita will always be one power less than the denominator. So this will be a constant because the numerator is linear plus be also constant because it's in America is linear. So now we're gonna multiply every term by the common denominator which will be ex express one every time cake So obviously this is canceled here only access cancel And here express one's castle. So what do we have left? We have one because to eat X plus one plus BX. Now let's attempt to find A and B let x v zoo. So on the left hand side, we still have one zero plus one, but become zero just be zero. So that gives us equals to one next, the XP minus one so that my kids can go to zero. So one request to behave minus one last one. So this becomes you go. So that's not considered ready. Plus B comes when it's one. So be ease minus one. So our final answer would be one over X square plus X. This equals two a is one over X these managements of minus one over X plus one. Now, if we were to check this where the answer is correct, you just common base the right hand side here. Now I don't have space. I'm just gonna do it right over here. Common base will be Ex express one. So my left will be this and my right will be this. You can see that. Yeah, it is the left hand side over here, so answer is correct.

We want to break down this fraction into its partial fraction. But ask yourself three questions. First question is the fresher, improper, improper means the numerator, highest degree or highest orders. Highest power is greater equals to the denominator. Highest power in this case, No, you can see that the numerator has power Zero because the number and the denominator highest power is 20 is less than two. So this fraction is proper Now, In any case, if you have an improper fraction, please do long division first. Second, can the denominator be fact arise? Yes, in this case. So let's characterize it. They told the X you get two X plus one it are there repeater factors. No repeater factors will look something like this express with them. So in this case, no, we do not have a bitter in fact this year. So we are ready now to express diffraction into special session. So over X plus over to express one, the numerator will be one power less than the denominators. So in this case, here will be a constant number because the denominator is linear. Same thing for to express one's linear, so the numerator will be a constant be now multiply every term by the common denominator, which will be extra X plus one. You can see that things cancels up the axis. Cancel out here and for here to express one's cancel out. So we're left with one. It was to be 20 plus one. Let's be ex to find nb that x zero so that this part can be set to zero. So one equals two a zero plus one plus beatem zero Just be zero. So hour is one makes the XB minus, huh? So this part can become zero. So I have one is equals to a zero plus two times minus half. So I will be Will be minus two. So therefore, my past reflection East A over X is one over X plus B over twice plus one. So these managed to so managed to over to express one. Now let's check about Answer is correct. So the right hand side, let's Coleman basic. So I'll get two x plus one minutes to X. So these cancels. I'll get one over x two x plus one, Which is this? So my answer is correct

Rational expression. One upon X square. Negative one. Now you can see that in the denominator. We have X square, negative one. So we have to find the factor off the denominator, which is access Squire. Negative one equal toe eggs plus one time X Negative one. Now we put that factor in the denominator and we get Guan upon X plus one time eggs. Negative one. Now we have to write form off decomposition. So we know that a a bone X plus one true and second term is plus B upon xnegative one. Now you can write like that And after that we have to take LCD, which is least common denominator. So we take And here one of them eggs plus one time eggs. Negative one equal toe eg a time Xnegative one plus b positive one X plus one upon X plus one time eggs. Negative one. So here you can see that we have to multiply X plus one time X minus one from the both sides. Then we get e time x 91 plus B time. Eggs plus one equal. Do one. No, we put X equal toe one. Then you can see that a drum will be zero and we get the terms so you can see that here when we would execute toe one. Then we get e time zero plus be one plus one equal to one. So we get be equal to one. Upon to now we put execute toe negative one. Then we get mhm time. Negative one negative one plus be time negative one plus one equal to one. Then we get negative two times a equal toe work It is B and after simplifying this, we get negative one upon to so a equal to negative one off to Now we put the value of A and B in this expression and we get one upon actress choir Negative one equal toe Negative one upon two times X plus one plus one of +12 time Xnegative one. Now you can take GCS which is one upon too. So our final answer aids one upon to bracket one upon eggs Negative one negative one upon X plus one So it is all final answer


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