2. The cubic formula
Quando chel cubo con le cose appresso When the cube with the cose beside it,
Se agguaglia `a qualche numero discreto equates itself to some other whole number,
Trouan dui altri differenti in esso. . .. find two others, of which it is the difference. . . .
First three lines of a 25 line poem in which Tartaglia (1539) described the cubic formula. Translation by
Friedrich Katscher (2011).
Let ω=−1+√−3
2. Remember that
ω2+ω+ 1 = 0 and ω3= 1.
Let x3−e1x2+e2x−e3be a cubic with roots r1,r2,r3, which we want to find. So we have
e1=r1+r2+r3e2=r1r2+r1r3+r2r3e3=r1r2r3.
Define the following quantities:
(1) s1=r1+ωr2+ω2r3
s2=r1+ω2r2+ωr3
Define the quadratic polynomial
(2) y2−f1y+f2:= (y−s3
1)(y−s3
2).
The cubic formula can be described algorithmically as follows:
Step 1: Find the coefficients f1and f2of the quadratic (2).
Step 2: Use the quadratic formula to solve for s3
1and s3
2.
Step 3: Take cube roots to find s1and s2.
Step 4: Solve the linear equations (1), together with the equation e1=r1+r2+r3, to find r1,r2and r3.
Details of Step 1
Problem 2.1. Consider permuting the roots r1,r2and r3. (For example, you might switch r2and r3, or
cycle r1→r2→r3→r1.) How are the following quantities affected?
(1) e1,e2and e3
(2) s1and s2.
(3) s3
1and s3
2.
(4) f1and f2.
Problem 2.2. Find formulas, in terms of e1,e2and e3, for the following quantities. I have prepared a
cheat sheet of useful formulas (next page).
(1) s1s2
(2) f1
(3) f2
Details of Step 4
Problem 2.3. Solve the linear equations (1), together with the equation e1=r1+r2+r3, to find r1,r2
and r3in terms of e1,s1and s2.
One last detail
As I described the algorithm, one takes cube roots twice, once to compute s1and once to compute s2. In
fact, one should only take one cube root, and then compute s2by the formula s2=s1s2
s1. (Now you know
why I had you compute s1s2in Problem 2.2.) Otherwise, out of the 9 choices of which cube root to take,
only 3 of the combinations will give correct solutions.
