Descartes’
method of solving quartic polynomials (polynomials of the 4th degree) in the
form 
has
four steps (Rubinstein, 4):
Make 
into a related ‘depressed’ quartic by plugging
in 
for 
. The ‘depressed’ quartic should now be of the
form 
. (Rubinstein, 4)
The real roots of this ‘depressed’
quartic are the intersection of the parabola 
and a circle that I will expound on in step 2.
(Rubinstein, 4)
To find the circle, start with the
‘depressed’ quartic that you found in step 1 and set equal to zero, 
. Expand the term 
as
. Our depressed quartic now looks like, 
. We will now substitute in for the first two terms. We now have 
. Subtract the r over: 
. Use complete the square method to write as
circle equation, 
. (Rubinstein, 4)
Graph the circle you found in step
two and the parabola, on the same coordinate plane. The x-coordinates of the intersections of the
circle with the parabola, . We graph the parabola because we let in our derivation of the circle. (Rubinstein, 4)
But what do the roots of the ‘depressed’
quartic tell us about the original polynomial? When I first read how Descartes’
found the roots to quartics, I said to myself, “So
what? So he found how to solve for the
roots of ‘depressed’ quartics, but what does this
tell me about the roots of the original polynomial?” Then I realized it was more trivial than I
was trying to make it. After you find
the real roots to the quartic function, simply substitute your roots into . (Remember, this was what we plugged in for x
in step 1 to get our ‘depressed’ quartic). (Rubinstein, 4)
See it in action!
Use the applet below to see the x-values of the intersections of the circle and the parabola. Compare with the x-values of the roots of g(x).
Move p (the coefficient on x^2 in g(x)) and see how it effects the circle.
Move q (the coeffiecient on x in g(x)) and see how it effects the circle.
Move r (the constant of g(x)) and see how it effects the circle.
Click the checkbox on g(x) to see the quartic and its roots.
(Rubinstein, 4)
Click here for a challenge!