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# Complex Partial Fractions

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### Roots and Coefficients

### Target Six

### 8 Methods for Three by One

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Age 16 to 18

Challenge Level

Find real constants $A, B$ and $C$ and complex constants $D$ and $E$ such that $${10x^2-2x+4\over x^3 + x} = {A\over x} +{Bx+C\over x^2+1} = {A\over x} + {D\over x-i} + {E\over x+i}.$$

NOTES AND BACKGROUND

*This problem gives an example where a rational function can be reduced to a sum of linear partial fractions IF we allow ourselves to use complex numbers. It turns out that this is always possible! This is of use in more advanced university-level applications of integration and analysis of series.*

If xyz = 1 and x+y+z =1/x + 1/y + 1/z show that at least one of these numbers must be 1. Now for the complexity! When are the other numbers real and when are they complex?

Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.

This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different? Which do you like best?