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

56

THEORY OF EQUATIONS

Ex. 3. Apply Stumrl's Theorell to 2 x5 + 7 x + 8 x + 2 x2 - 2 x - 1 = 0
We find fI'(X) = 10X4 + 28x3 + 24 2+4 x - 2,
f (z) = 3 + 3 x2 + 3 X + 1.
Here f2(x) is found to be the II. C. F. of f (x) and f'l(); hence - 1 is a
quadruple root. For x +a=, the functions f(x), f'(x), f:(x) yield the
signs + + +; for x = -o they yield -- +-. Hence there are two
distinct real roots, and all the roots are real.
Ex. 4. Show that all the roots of x + x3 -x2 - 2x + 4 0 are
imaginary.
Ex. 5. Required the number and situation of the real roots of
2x - 11 x2 + 8x- 16 0,
x3 + 11 x2 - 102 x +- 181 = 0,
x5 - 36 x3 + 72 x - 37 x + 72 = 0.
50. Nature of the Roots of the Quartic. In the study of the
nature of the roots of the cubic equation we began in ~ 35 by
deducing the " equation of squared differences of the roots of
the cubic." Then, in ~ 36, we used this transformed equation
in the discussion of the roots of the given cubic. The same
mode of procedure might be adopted in the study of the roots
of the quartic equation. But the formation of the " equation
of squared differences of the roots" is laborious, and we prefer
to begin the discussion by applying Sturm's Theorem to the
quartic with its second term removed.
If we transform the general quartic
box4 + 4 bx3+ 6 bx2 + 4 bx + b4 = 0, I
into a new equation, deprived of its second term and with coefficients integral in form, we obtain, as in ~ 34,
y4 +6 Hy24 + Gy + boI-3 H' =0, II
where y = bo + b,
H_ bob,,- b 2
G bo2b3- 3 boblb, + 2 b3,
I- bob4 - 4 bb3 - 3 b22.