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Girard Relations


Albert Girard, Belgian mathematician born in the year 1595, in his studies established mathematical formulas that relate the coefficients and roots of an algebraic equation.

Consider the polynomial of 2nd degree:

P (x) = ax² + bx + c, with The0,

whose roots are . Note that:



Equating the coefficients:

 

Consider the polynomial of 3rd grade P (x) = ax³ + bx² + cx + d, with The0whose roots are . Note that:


Equating the coefficients:

By reasoning analogously to the previous ones, we find the relations to an algebraic equation of any degree no:

Example
Solve the equation , knowing that one root equals the sum of the other two ():

Resolution

Girard Ratio:

Let's sharex³ - 10x² + 31x - 30 per (x - 5) to find the other roots:

Q (x) = x² - 5x + 6

So the solution set of the equation é:

S = {2,3,5}
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