An introduction to the modern theory of equations, by Florian Cajori.

ELEMENTARY TRANSFORMATIONS OF EQUATIONS 41

Ex. 1. Find the equation of squared differences of the roots of the
cubic x3-+ 3 x2 - 3 - 1 = 0.
Here bo = 1, b = 1, b2 -1, b3 =-1. Hence G = 4 and lz= -- 2.
The required equation is z3 - 36 z2 + 324 z - 432 = 0.
Ex. 2. The cubic in the previous example is a reciprocal equation.
Solve it, find the values of the squared differences of the roots, and see
whether they are really roots of the equation of squared differences.
The reciprocal equation of the standard form, obtained from the above,
is x2 + 4 x + 1 = 0. The roots of the given cubic are 1, -2 ~ x/3; their
squared differences are 12, 12 ~ 6V/3. Dividing the left member of the
transformed cubic by z - 12, thus,
1 - 36 + 324 - 432 112
+ 12 - 288 + 432
- 24 + 36 + 0
we see, by ~ 4, that 12 is a root. The depressed equation, z2- 24 z+36 =0,
is satisfied by z = 12 -i 6V/3.
Ex. 3. Find the equation of squared differences of the roots of the
cubic x3 + x2 - x - 1 =0.
The required equation is z - 8 z2 + 16 z = 0. What inference can be
drawn with respect to the roots of the given cubic from the fact that - 0
is a root of the transformed cubic?
Ex. 4. Find the equation of the squared differences of the roots of
x3 + 3 x + 2=0. Ans. Z3 + 18 2 + 81 + 216 = 0.
It is important to observe that, since the last term + 216 is positive,
and is equal to minus the product of the roots, at least one of the three
values of z must be negative. Now if the roots of the given cubic are all
real, then the squares of their differences must be positive, and all the
values of z must be positive. A negative value of z can be obtained only
when the given cubic has two imaginary roots. Hence xa + 3 x + 2 = 0
has two imaginary roots. Verify this by Descartes' Rule of Signs.
Ex. 5. Find the equation of the squared differences of the roots of
X3 + x2 + 5 x - 16 = 0.
The process is easier if we first transform the cubic to another whose
second term is wanting.
36. Criteria of the Nature of the Roots of the Cubic. We proceed to discuss the nature of the roots of the general cubic I in:,5, )witll the he1lp of the " equation of squared differences " V.