Partial Fractions

[Zaqiqu]

Above is a screenshot from an online partial fractions calculator. They give the solution, but it doesn't mesh with what I know. I give my own work below, which I understood immediately to be wrong. I've tried three times, each ending up at the same place. The problem is not with my arithmetic, but with my technique.

The problem I seem to be having is when I get to
A(x^2)(x+1) + B(x)(x+1) + C(x)(x^2)
A(x^3+x^2) + B(x^2+x) + Cx^3
(A + C)x^3 + (A + B)x^2 + Bx

I tried turning that into a matrix with A+C=0 (since X^3=0), but that was obviously not correct. What do I do with these cubes?

 

[Deleted member 4993]

Above is a screenshot from an online partial fractions calculator. They give the solution, but it doesn't mesh with what I know. I give my own work below, which I understood immediately to be wrong. I've tried three times, each ending up at the same place. The problem is not with my arithmetic, but with my technique.

The problem I seem to be having is when I get to
A(x^2)(x+1) + B(x)(x+1) + C(x)(x^2)
A(x^3+x^2) + B(x^2+x) + Cx^3
(A + C)x^3 + (A + B)x^2 + Bx

I tried turning that into a matrix with A+C=0 (since X^3=0), but that was obviously not correct. What do I do with these cubes?

There is problem with your arithmetic...

\(\displaystyle \displaystyle{\frac{A}{x} + \frac{B}{x^2} + \frac{C}{x+1} \ = \ \frac{x^2+1}{x^3+x^2}}\)

\(\displaystyle \displaystyle{\frac{Ax(x+1)}{x^2(x+1)} + \frac{B(x+1)}{x^2(x+1)} + \frac{Cx^2}{x^2(x+1)}\ = \ \frac{x^2+1}{x^3+x^2}}\)

No cubic terms in the numerator ... continue...

 

[Zaqiqu]

There is problem with your arithmetic...

\(\displaystyle \displaystyle{\frac{A}{x} + \frac{B}{x^2} + \frac{C}{x+1} \ = \ \frac{x^2+1}{x^3+x^2}}\)

\(\displaystyle \displaystyle{\frac{Ax(x+1)}{x^2(x+1)} + \frac{B(x+1)}{x^2(x+1)} + \frac{Cx^2}{x^2(x+1)}\ = \ \frac{x^2+1}{x^3+x^2}}\)

No cubic terms in the numerator ... continue...

I couldn't seem to figure out why the A has only the variable x next to it, rather than x^2. I don't think it was an arithmetic problem on my part. My problem was my rote following of the given procedure, rather than asking why is this procedure valid.
I had no problems at all with the first 20 questions in Stewart's Precalculus 10.7. I think I just figured it out. You wrote it out in a more clear way than Stewart does in the whole chapter. He definitely could have used your clarity in explaining himself.

I'm almost done Stewart's Precalculus, and I plan on starting a calculus textbook immediately thereafter. I have access to three:
Larson's Calculus: Early Transcendentals
Stewart's Calculus 7E
Hughes-Hallett Calculus: Single and Multivariable

I had intended on using Stewart's, since I am doing his precalculus course. Are you familiar with any of these? Are they any good? I don't care to be "spoon-fed", but I want something that provides thorough explanations, not something that'll leave me scratching my head for two hours because the author omitted useful information.

 

[Deleted member 4993]

I couldn't seem to figure out why the A has only the variable x next to it, rather than x^2. I don't think it was an arithmetic problem on my part. My problem was my rote following of the given procedure, rather than asking why is this procedure valid.
I had no problems at all with the first 20 questions in Stewart's Precalculus 10.7. I think I just figured it out. You wrote it out in a more clear way than Stewart does in the whole chapter. He definitely could have used your clarity in explaining himself.

I'm almost done Stewart's Precalculus, and I plan on starting a calculus textbook immediately thereafter. I have access to three:
Larson's Calculus: Early Transcendentals
Stewart's Calculus 7E
Hughes-Hallett Calculus: Single and Multivariable

I had intended on using Stewart's, since I am doing his precalculus course. Are you familiar with any of these? Are they any good? I don't care to be "spoon-fed", but I want something that provides thorough explanations, not something that'll leave me scratching my head for two hours because the author omitted useful information.

I am an engineering instructor - not a math instructor. So I don't think I am qualified to provide you with a comparative assessment of those books. Our Mathematics department uses the Larson's book.