- #1
MathewsMD
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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 A1 and An in this case.
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 A1 and An 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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