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Decomposing Fractions [Solved!]

My question

In decomposing the fraction of Chapter 11 Integration example 3 b, we are trying to figure out if there is a way to find the coefficients on `x^3` without expanding.

Relevant page

11. Integration By Partial Fractions

What I've done so far

We expanded: `A(x^3 - x^2 - x - 1) +` ` ... +D(x^3 + x^2 + x- x^2 - x - 1)`

X

In decomposing the fraction of Chapter 11 Integration example 3 b, we are trying to figure out if there is a way to find the coefficients on `x^3` without expanding.
Relevant page

<a href="https://www.intmath.com/methods-integration/11-integration-partial-fractions.php">11. Integration By Partial Fractions</a>

What I've done so far

We expanded:  `A(x^3 - x^2 - x - 1) +` ` ... +D(x^3 + x^2 + x- x^2 - x - 1)`

Re: Decomposing Fractions

I haven't heard of any other techniques for finding partial fractions outside of the one explained on the page you came from. (Paul's Online Notes does it the same way.)

I'm not quite sure what you are doing so far. What you have is equivalent to the following, and I'm not sure where it comes from:

`A(x^3 - x^2 - x - 1) + ... +D(x^3-1)`

X

I haven't heard of any other techniques for finding partial fractions outside of the one explained on the page you came from. (<a href="http://tutorial.math.lamar.edu/Classes/CalcII/PartialFractions.aspx">Paul's Online Notes</a> does it the same way.)

I'm not quite sure what you are doing so far. What you have is equivalent to the following, and I'm not sure where it comes from:

`A(x^3 - x^2 - x - 1) + ... +D(x^3-1)`

Re: Decomposing Fractions

It was obtained by fully expanding `(x-1)^2(x-1)` = `x^3+x^2-2x^2-2x+x+1`. The same was done with `(x-1)^3.

But I now realize when seeking the coefficients for a term then only operate on that term. It is not necessary to fully expand and find the coefficient of each term if it is not needed.

X

It was obtained by fully expanding `(x-1)^2(x-1)` = `x^3+x^2-2x^2-2x+x+1`.  The same was done with `(x-1)^3.  

But I now realize when seeking the coefficients for a term then only operate on that term. It is not necessary to fully expand and find the coefficient of each term if it is not needed.

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