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I know how to do these problems but this one is giving me some trouble: Expand into partial fractions

$$\frac5{(x+1)(x^2-1)}$$

*Should I break it down into:
1) $\frac A{x+1}+\frac B{x-1}+\frac C{x+1}$ or
2) $\frac A{x+1}+\frac B{(x+1)^2}+\frac C{x-1}$

I tried it both ways and came up with different possibilities for the constraints:
If Broken Down Like #1 from Above $$A+B+C=0,\hspace{10pt} 2B = 0,\hspace{10pt} A+B-C=5$$

If Broken Down Like #2 from Above $$A+C=0,\hspace{10pt} B+2C=0,\hspace{10pt} A+B-C=5$$

  • 0
    See [http://www.purplemath.com/modules/partfrac.htm](http://www.purplemath.com/modules/partfrac.htm).2012-11-26
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    You need to go to page 2 of the link @MhenniBenghorbal provided above, to see how to deal with repeated factors in the denominator.2012-11-26
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    I haven't gone through the calculation, but something about your first attempt doesn't look right. The expression in 1) is symmetric wrt $A$ and $C$, so the resulting equations should have the same symmetry. In fact your equations, if set up right, should look like three equations in two unknowns, $A+C$ and $B$, and so overdetermined. Which gives you a strong hint that 1) is not going to work.2012-11-26
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    @HaraldHanche-Olsen is right. We have for 1) $A(x^2-1) + B(x+1)^2 +C(x^2-1) = 5$, giving $A+B+C = 5$, $2B = 0$, $-A + B -C = 0$. (so $A+C = 5$ and $A+C = B = 0$.2012-11-26
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    I agree, #1 is wrong so i went with option 2.. However, i can't figure out my constraints to solve for A, B & C..any help?2012-11-26
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    The simplest way is to multiply the equation $\dfrac{5}{(x+1)(x^2-1)} = \dfrac{A}{x+1} + \dfrac{B}{(x+1)^2} + \dfrac{C}{x-1}$ by the denominator $(x+1)(x^2-1) = (x+1)^2(x-1)$ and look at the coefficients of each power of $x$.2012-11-26
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    A cleverer way is (1) multiply by $x-1$ and substitute $x=1$, getting the value of $C$. (2) multiply by $(x+1)^2$ and substitute $x=-1$, getting the value of $B$. (3) after plugging in those values for $B$ and $C$, take the difference between the two sides, simplify as see what $A$ must be.2012-11-26

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