Find the partial fraction decomposition for each rational express...

Question

Find the partial fraction decomposition for each rational expression. 5-2x / (x^2 + 2)(x - 1)

Accepted Answer

Hello everybody. I hope you're doing all right. Today, today we're going to be looking at this question that states where the following rational expression applied partial fraction decomposition, where our expression is a fraction with seven minus three X in the numerator. And that is divided by open parentheses, X squared plus seven closed parentheses, multiplying open parentheses, X minus three closed parentheses in the denominator. So that multiplication and entire multiplication will take place in the denominator. Now the answer choices A through D are a set of two fractions, adding each other with answer choice. A having the first fraction being negative one in the deno in the numerator divided by eight multiplying and open parentheses. X minus three, close parentheses in the denominator plus the second fraction which will be X minus 21 in the numerator. And the denominator will be eight multiplying open parentheses. X squared plus seven closed parentheses as, as being is the same set of two fractions except the first fraction has a positive one in the numerator. C once again is the same set of two fractions except now we have an X plus 21 in the numerator of the second fraction and answer choice D once again, the same two fractions except the first fraction will have a positive one in the numerator. And the second fraction will have an X plus 21 in the numerator. No. When tagging this question, I have to ask myself what is partial fraction decomposition? So what they give us is a, they give us a fraction. That's our expression. We have a complicated fraction with a numerator with an X and the denominator with two parentheses, multiplying each other with an X squared and another X. So a decomposition is basically just trying to figure out what to, how we can turn this one fraction into multiple fractions that when multiplied will give me this one fraction once again. So we're trying to decompose it and a general formula for that. So in this case, we have one of our parentheses is a quadratic in a quadratic form. So we have are expression equal to and will have a fraction with a multiplying X. So A X plus B in the numerator of the fraction, and that fraction will be divided by X squared plus seven. So here we're trying to solve for A and for B because uh as I, as I mentioned, we're trying to decompose the fraction. So we know that our denominator will be one of the parentheses in our original expression. So X squared plus seven in this case, and we're trying to solve for what the numerator would be in this fraction. And that fraction will be plus another fraction which would be C in the numerator. And the denominator will have the other parentheses in this case X minus three. So this is how it would look like decomposed our expression decomposed. So now we have to figure out what A B and C would be. So let's start doing that. The first thing we have to do is multiply both sides of our expression by X squared plus seven and X minus three. So I'll draw a parentheses covering our entire expression. And outside of that parentheses, I'll write the X squared plus seven and X minus three since we're gonna be mo multiplying that and we're multiplying both sides by that. So I'll write it on the right side as well. So we'll have X squared plus seven multiplying X minus three and that will be multiplying both sides of our expression. So if we do look at the left hand side, now, the X squared plus seven and the X minus three will cancel in the new, in the denominator or fraction. So we'll have seven minus three X now and that'll be equal to. Now, we look at the first term of our right hand side. So our first fraction here, which was A X plus B divided by X word plus seven, the X squared plus seven will cancel and will be left over with A X plus B multiplying X minus three since the X minus three won't cancel in this fraction. And we'll have that plus. And now our second term was C divided by X minus three and that X minus three will cancel with the X minus three. We're multiplying the entire equation by. So it will be left over with C multiplying X squared plus seven. And now this will be our expression now, so we'll have seven minus three, X equals open parentheses. A X plus B closed parentheses, multiplying and open parentheses, X minus three, closed parentheses plus C multiplying and open parentheses, X squared plus seven closed parentheses. Now to start solving for the variables that we want. So A B and C there's multiple routes that we can take, so we can distribute everything, use foil and kind of distribute everything and then write a system of equations and start canceling out variables solving for the others. However, we can also try out strategic values for X that will help us solve. So what I mean by that is in this case, let's say we let X equal three. And why did I pick three? Well, if I put in a three for the X, my parentheses that had X minus three, now we have three minus three which will be zero and that parenthesis was multiplying A X plus B. So zero, multiplying by that will be zero and we eliminate variable A and B and will only be left with the variable C which we can then just solve for if we plug in three for the X. So that's exactly what we'll do. So now we have seven minus three, multiplying three, it equals, we'll have a multiply it by zero plus B was flying or not multiplied by zero, but a multiplied by three. Since that is R X plus B multiplying a three minus three in the other parentheses, that'll be plus C multiplying an open parentheses three squared plus seven. And like I said, three minus three will be zero and zero, multiplied by the parentheses that had A X plus B will cancel, that will also be zero. So now we can go ahead and distribute our values. So we'll have seven minus three, multiplied by three is seven minus nine. And that'll be equal to C multiplying a three squared which is nine plus seven. And I'm just gonna move my work over a bit to have a bit of more room to work with. So now I can go ahead and distribute. So I have seven minus nine which is negative two. So I have negative two equals C multi or in the inside parentheses. Now we have nine plus seven which is 16 and 16, multiplied by C will be 16 C. And now if we divide both sides by negative two, we'll have that C equals negative one divided by eight. So that is our first a solution. For this. So now we can go ahead and continue solving for the other variables. So now is when I'll go ahead and do the foiling that I mentioned earlier. So I'll go back to my equation that I originally had, which was seven minus three X is equal to open parentheses. A X plus B closed parentheses, multiplying an open parentheses, X minus three plus C multiplying open parentheses, X squared plus seven closed parentheses. And now I can go ahead and distribute everything. So we'll have seven minus three X equals. We have a X multiplying an X. So that'll be A X squared. And then we'll have a X multiplying a negative three which will be minus three A X. Then we have a positive B multiplying an A positive X. So B plus B X. And finally, we have the positive B once by a negative three which will give us minus three B and then we can have the other distributor. So we'll have plus and then C multiplied by X squared, which will be C X squared and finally, C multiplied by a positive seven which will be plus seven C. And I'm gonna rearrange our fraction or our equation to have on the left hand side, I'll have negative three X first. So I'll rearrange it to have negative three X plus seven equals. And then I'm gonna, every term that has an X squared, I'll put it first and then any term that has an X. And then finally, any term that doesn't have any XS, so I'll have A X squared plus C X squared minus three A X plus B X plus B X minus three B plus seven C. So just going to repeat that we'll have negative three X plus seven equals A X squared plus C X squared minus three A X plus B X minus three B plus seven C. So now we can compare both sides because on the right hand, on the left hand side, we don't have any X squared. That means on the right hand side, our X squared terms must add up to zero because the terms on the left hand side must match the terms on the right hand side. So that means that A plus C equals zero because again, we have no X squared term on the left hand side. So this means that A equals negative C. So A equals positive one divided by A and now we can continue. Now we have C and A and we can go ahead and continue so we can use that same logic with other values here. And once again, I'm going to move my work a bit to have more room. So using the same logic of matching the terms on the right hand side and the left hand side, we see that our values with or, or the value with no X on the left hand side is A seven, we have negative three X plus seven. So that seven is the value with no, the term with no X attached to it. And now on the right hand side, the terms with no X attached to them to them is negative three B plus seven C. So we can have that seven C minus three B equals seven. So seven C minus three B is the term we had on the right hand side. And that must equal the term on the left hand side, which is seven. So now we can plug in the scene and we have seven multiplied by a negative one divided by A minus three B equals seven seven. Multiplying a negative one divided by eight will be a negative seven divided by eight minus three B equals seven. Now I can move the negative seven divided by eight from the left hand side to the right hand side. And we'll have negative three B equals seven plus seven divided by eight. And now we can turn our seven into a fraction with a denominator of eight. So we can add this up. So we'll have negative three B equals 56 divided by eight plus seven divided by eight. And now we have, we can simplify that to have negative three B equals 56 plus seven is 63. So we'll have 63 divided by eight. And now if we move the negative three over from the left hand side to the right hand side by dividing, we'll have that B is equal to 63 divided by eight, multiplied by negative three in the denominator. That will be equal to 63 divided by negative which means that B is equal to negative 21 divided by eight. If we simplify that fraction by dividing both the numerator and the denominator by three. So that will be our answer for me. Now, what you would do here, you have your A value, your B value and your C value. And you can just compare that or, or plug that into our equation that we had initially. So we had that our fraction or our ex our rational expression was equal to A X plus B divided by X squared plus seven plus C divided by X minus three. So if we plug in the C, which was negative one, divided by eight, we'll have a negative one divided by eight multiplying and X minus three. So right away, we can eliminate answer choice B and answer choice D because they have a positive one for that fraction. And next, we know that we'll have a negative 21 because of R B. So we can eliminate for choice C which has X plus 21. So we're gonna be left with answer choice A which is the set of two fractions that was negative one in the numerator divided by eight multiplying an X minus three. In the denominator plus another fraction with X minus 21 in the numerator divided by eight multiplying an X squared plus seven in the denominator. So I hope that was helpful. And until next time.

Find the partial fraction decomposition for each rational expression. 5-2x / (x^2 + 2)(x - 1)

Key Concepts

Rational Expressions

Partial Fraction Decomposition

Factoring Polynomials

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