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Write the partial fraction decomposition for 10 (1 + )(c - 2)2...

Question

Write the partial fraction decomposition for 10 (1 + )(c - 2)2

Write the partial fraction decomposition for 10 (1 + )(c - 2)2



Answers

write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $$\frac{11 x-10}{(x-2)(x+1)}$$

Okay, Uh, decomposing. This first recognized that the denominator factors into X times X minus one. So the form for our decomposition will be something over X, plus something else over X minus one. So that's how this would break down. And then we would multiply everything by the common denominator of the X squared minus X, in which case that's going to just get rid of all our denominators. And over here on this side, each of these fractions the numerator gets multiplied by the other denominator because in this case, the X would divide out. And so we have a times the X minus one. So 11 X minus 10 equals a times the X minus one. So a X minus a loss and here will have be times the x p x. And now we can equate the coefficients of the terms on each side. So a pause be past equals 11. And for our Constance, we have negative. 10 has to equal negative a. So the solution to this system of equations A is 10 and B equals one. And so that means our decomposition will be 10 over x plus one over X minus one

In the problem we have been given then X plus three upon X into X-us one. These equals airborne, X plus be upon X plus one. Now this is written as 10 X plus three that equals two And works plus one plus VX. Now this is three that equals into one are equal three. Now further minus Tan plus three equals B Or -7 equals b. Or be equal. This is -7. So This cancels each other, hence be equal seven. Therefore, partial fractions can be written us three upon X Plus seven a bond X-us one. So we have this is our answer.

Getting for this problem leads factor The denominator We'll have an X Times X minus one so they probably formed would be in a over and X plus B over an excellent this one.

We have to find the partial fraction decomposition of 10 minus X divided by X square last 10 x last 25. Right. So to do that, begin divide this into factors that will be well X plus five. Hold square. This is just express our whole square. So what do we do with this? Is 10 by X plus five whole square eyes equal to a X plus five. Let's be express five whole square. Let's assume that these two fractions is what Ah, yeah, assuming is that these two fractions is what this whole numbers made up off, right? No. So to find at, uh, to find A and B well, these air that's partial fractions right on to find a and B do is remove the denominators and multiply it here. So you get what you get is 10 off minus X. So 10 minus X is equal to a X plus five. Let's be so now to get the value V make X is included minus five. Get value fee. Uh, it should be 10 plus five years since his minus Here, it's gonna be plus, is equal to be so 15 now to find the value. Fay, you're gonna have 10 Minus X is equal to a X plus five plus 15. Right. They give X any arbitrary around you. Let's just say zero it's gonna be 10 is equally thio five a less 15 Just gonna be minus five is equal Thio five day it's gonna be a zero to minus one. So that's how you find the values. So the parts production would be minus one X plus five plus 15 x plus five old square.


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