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http://en.wikipedia.org/wiki/Partial_fraction_decomposition

In general, if you have a proper rational function, then:

if ## R(x) = \frac {P(x)}{Q(x)} ## and ## Q(x) = (mx + b)^n ... (ax^2 + bx + c)^p ## where ##Q(x)## is composed of distinct linear powers and/or distinct irreducible quadratic powers.

I had two questions:

1) Can we actually reduce every polynomial of powers greater than 2 to either a distinct linear or quadratic power? I may not be thinking correctly, but does anyone know of a proof they could link me to? For a an expression like ##x^{12} - πx^3 - 542.43x + 21## I am having trouble simplifying this to an irreducible quadratic or linear factor.

2) Why can we show that ## R(x) = \frac {A}{mx+b} + \frac {B}{(mx+b)^2}... \frac {C}{(mx+b)^2} + \frac {Lx + D}{ax^2 + bx + c} + \frac {Mx + E}{(ax^2 + bx + c)^2}...+\frac {Nx + F}{(ax^2 + bx + c)^p} ##

For each distinct linear and irreducible quadratic power, why can we not just show this as:

## R(x) = \frac {A}{(mx+b)^n} + \frac {Lx + D}{(ax^2 + bx + c)^p} ##

I know the above is incorrect, but I'm wondering how it was proved that we must add a power to the denominator for each subsequent fraction in the series.

Also, I've seen it written as ## A_1 + A_2 +....+ A_n## instead of ## A + B +....+ C## and I was wondering if there's any relationship between A

I've looked online but have not found the proof(s) that I'm looking for.

Any clarification would be great!

In general, if you have a proper rational function, then:

if ## R(x) = \frac {P(x)}{Q(x)} ## and ## Q(x) = (mx + b)^n ... (ax^2 + bx + c)^p ## where ##Q(x)## is composed of distinct linear powers and/or distinct irreducible quadratic powers.

I had two questions:

1) Can we actually reduce every polynomial of powers greater than 2 to either a distinct linear or quadratic power? I may not be thinking correctly, but does anyone know of a proof they could link me to? For a an expression like ##x^{12} - πx^3 - 542.43x + 21## I am having trouble simplifying this to an irreducible quadratic or linear factor.

2) Why can we show that ## R(x) = \frac {A}{mx+b} + \frac {B}{(mx+b)^2}... \frac {C}{(mx+b)^2} + \frac {Lx + D}{ax^2 + bx + c} + \frac {Mx + E}{(ax^2 + bx + c)^2}...+\frac {Nx + F}{(ax^2 + bx + c)^p} ##

For each distinct linear and irreducible quadratic power, why can we not just show this as:

## R(x) = \frac {A}{(mx+b)^n} + \frac {Lx + D}{(ax^2 + bx + c)^p} ##

I know the above is incorrect, but I'm wondering how it was proved that we must add a power to the denominator for each subsequent fraction in the series.

Also, I've seen it written as ## A_1 + A_2 +....+ A_n## instead of ## A + B +....+ C## and I was wondering if there's any relationship between A

_{1}and A_{n}in this case.I've looked online but have not found the proof(s) that I'm looking for.

Any clarification would be great!

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