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Write the partial fraction decomposition for the rational expression. Check your result algebraically by combining fractions, and check your result graphically by u...

Question

Write the partial fraction decomposition for the rational expression. Check your result algebraically by combining fractions, and check your result graphically by using a graphing utility tograph the rational expression and the partial fractions in the same viewing window. $$ rac{x^{2}+12 x-9}{x^{3}-9 x}$$

Write the partial fraction decomposition for the rational expression. Check your result algebraically by combining fractions, and check your result graphically by using a graphing utility to graph the rational expression and the partial fractions in the same viewing window. $$\frac{x^{2}+12 x-9}{x^{3}-9 x}$$



Answers

Partial Fraction Decomposition Find the partial fraction decomposition of the rational function.
$$\frac{9 x^{2}-9 x+6}{2 x^{3}-x^{2}-8 x+4}$$

In order to find the partial fractions decomposition of the following rational expression, we must first look at the denominator. Now, here we could see that X squared is common between X of the port minus nine exports so we could factor in ah X squared out of extra fourth minus nine experts. So we taken X squared out. We're left with X squared minus not now X squared minus nine Can still be factored X where my science X minus three times x plus ter so we could write Rewrite This is two X plus three over X squared times X minus three X plus three. Now we're ready to start writing our partial fraction decomposition. So if you notice X squared is repeating. So we're gonna write some number A over X plus some number B over X where plus now we ever to non repeating functions, plus some number C over X minus three plus some number D over X plus. OK, now we're ready to try to solve for ABC. Indeed, let's multiply both sides of the equal signed by the common denominator X squared times X squared minus nine and the result will be two X plus three is equal to a X, so we'll multiply. X squared minus nine is X minus three times X plus three plus B times X minus three X plus three. Who are C X squared, X plus three plus de X squared X minus three. Now let's expand. Expand everything on the right. Multiply everything out and then group are like terms, and when you do that, you're left with the identity. Two X plus three is equal to a plus. C plus D. Ex queued waas plus three C was, uh minus was three C minus three D X squared, plus negative nine a axe plus negative nine. Be okay. Now that we approve dollar terms, let's equate the coefficients on both sides of the equal sides. Well, there's no execute on the laugh so we can write a plus B. A plus c plus de is equal is your There's no X Square to Grade B plus three CVE minus three D is equal to zero. There is another two exes on the left, so we could say we were right. That negative nine a will equal to. Likewise, there's a three here. Negative nine b that's what will correspond to that. So we could say that negative nine b is equal to three. OK, now we can saw Let's look at the, uh and be first. Well, if Negative nine A equals two, we divide both sides by negative nine. We're gonna be left with a equals negative, too nice. And for beef, we divide both sides by negative nine. We're gonna get negative three over none, which equals negative 1/3. Now that we have A and B, let's offer, See indeed. Well, let's look at this equation. So if a equals negative 1/3 C plus D will equal 1/3 we add a, which is Nate. We add us not negative. What their A will be negative two nights. So if a is negative two nights C plus D will equal to nice. What about this equation? If B is equal to negative 1/3 and three C minus three D is equal to 1/3. Now let's try solving these two equations on Teasley. Let's divide this equation by three were divided out by three. We're left with C minus. D is equal to one night. And now let's just rewrite this year because we're about to ab use to go c plus d is equal to night. Now we're gonna add we're gonna get rid of our deeds cell for C So this would mean that C plus C that's two C is equal to three night divide both sides by two and you're gonna get C is equal to 3/18 which is ableto warn six. Okay, now that we have a, B and C, let's solve for D Well, if C plus D equals two nights, then they're C plus D is equal to two nights. So that means +16 plus D is equal to two nights. So therefore, D is equal toe d will be equal to one E teeth 29 to minus 16 is 1 18 Now that we have a, B, C and D, we could finish writing are partial fraction decomposition of the rational expression. So a rational expression is two x plus three over X to the fourth minus nine X squared, so that will equal a is negative two nights. So that's gonna equal negative to ninth in your negative to ninth over. X plus B is negative 1/3. So plus negative 1/3 X square plus C is +16 of +16 x minus three plus de is 1 18 Suppose 1/18 X plus three and that is your parcel fraction decomposition of the rational expression in this problem.

Given expression is full exist Quiet plus her Phoenix minus nine, divided by X. Explain this one express they will help will be composed it in part infection. So we will use unknown constant so we can write it in my eggs plus B by X minus one. Let's see by express. Now our goal is to find a B and C for doing this living here. My life x X minus one expressed video boards. So what exists? Quiet. Bless 30 Knicks minus name divided by ex Explain this one. It's less three murder blood money. Thanks. Explain this. One express equals two A bikes plus be like playing this one Nos See you. Bye. X plus three Former Goodnight place X X minus one explicit. By solving this, we will get for expires. Let's 30. Next minus nine equals two e X minus one Explicit three. Let's be XX this seat. Let's see X X minus one. Unify this or what we will get. What if this quiet lets her be next? Minus nine equals do e exists Quiet minus six. Less three x minus t less Be X squared plus three x Let's see a crystalline and minus thanks. What exists? What if Lester Phoenix minus nine equals Stool? Yeah, Exit Square. Let's select minus he plus be excess. Where? Let's three X. Let's see Chris Plan My next thanks after solving it will get for it, despite with 13 x minus lane equals stoop. It is what it less to a X minus two years. Bless the X Square. Less TVX loves the excess. Where my next we've been here injured. We will get. What if this layer Let's have the next minus nine equals school. A letter B plus c exists player Bless weightless three B minus e. Thanks. Minus t is now if the polynomial czar saying then they're proficient are also Z So a plus B plus C very before through a bliss Leave me minus C equals soup, Herbie. Minus three years equal to minus nine. Pretty questionable one. The question. Number two. The question of breast trees. By solving these three questions, we will get a Z course quickly. Vehicle stroke, too, and C goods to mine. This one and the bunch of action is B by eggs blessed by X minus one, minus one by explosives. His

Given expression is nine x plus two. Divide by X minus two excess square less reckless. We need to be completed by perfection. For doing this, we do the non constant, so it is equal to a Did I explain this to nous? Be explicit. E Divide by exists player less to express. Now go live to find a B and C for doing this development. Replace X minus two. It's a squared plus two. It's the studio, both sides. So we will get nine expects you If I didn't buy X minus two X squared plus crew. It's less too more market led by X minus two X squared plus two X plus two equals Hey, divide makes minus two. Yes, The Ecstasy Divide by Thanks a squared plus two extra store for my good night Bye X minus two. It's a slayer Plus to express No before Okay, if this where Bless us too. Let's be explicit. If minor, we will solve it. Nine. You got it. Do it! Let the way. Let's vehicles place minus here. Yeah, my Now you re in invisible Big Nine Express school. Equal. Eagle it. Let's be it's this place. Let's do is minus to B plus C. Now a group on the owners are saying then their confusion arose in April. The acquittal. Dedeaux No, a minus movie. Let's be quick Line No. Eight. My life will be equal. What did the questionable one? Question number two. And it goes your number By holding the seat regions they will get Okay, be my and equal. But And the box is actually you divide by X minus school mind Look minus one goodbye. It was

Given expression is X Q minus four Dexter Square LS nine x minus. Faith divide by exits Glad minus two weeks, plus three hordes with you need to be compose it in part, a reflection for doing this. Maybe you the known constant, so it is equal. Do X plus B Divide by excess square minus two weeks plus three this see xsd divided by excess square minus two X plus three holes where now our goal is to find a B c N b. For doing this, we will my reply. Excess squared minus to express three holds Where do I would say so? It will get excuse minus 42 square. Bless my Yonex minus life. The right by It's a square minus two weeks plus three holds glad my divide by excess. Where minus two. Wicketless the words Where where you go? X Plus B Divide by X a square minus two X plus three. Bless, See explicit E Divide by X. Elsewhere, minus two recklessly was where all my replied by It's where, minus four weeks. Leslie I was with none of this old brick building X Q minus. Bordick's Square LS nine Knicks minus five equally toe Express. Be excess glad minus pro. X plus three Press See Explosive X Cube minus 40 square. Let's Nynex minus five equal x you minus through a excess square plus three x Let's be extra square minus movie X plus three B Let's see ex bless the now the very injured we will get. Excuse minus 40 square rest nine x minus. Fight equal a X cube Bless minus to a plus B Excess glass Yes, three A minus B plus C x plus three b Blessed be now the same than their confusions are also seem so equal to one minus to a plus. B equals minus four. Clean is minus. B plus C equals nine and three. Be bliss be equal to minus. Life would be good for number one question number. Do it with the number three and devotion number. By solving these four questions, we will get a 1 to 1. Be equal to minus toe, see equal toe, do and be equal toe one, and the party direction is X minus two. Divide by X a square minus two explosive C Bless two X plus one. Divide by excess square minus two X plus three holes where


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