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Roots and Coefficients

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?

Target Six

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

8 Methods for Three by One

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?

Complex Partial Fractions

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.