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

THEORY OF EQUATIONS

Ex. 4. Show that for the general equation f(x) = 0, the general form,
in terms of the coefficients, obtained for Zalta22 is the same as for the
quartic equation.
Ex. 5. Calculate Zcl3a2 for f(x)= 0 and from the result derive the
special value it assumes for the cubic.
Ex. 6. Calculate Ja12cc22a3 for the quintic equation. Is the result
the same for the general equation?
Ex. 7. Find the value of the symmetric function,c - /)2+ (/ - 7)2- (7 -( )2 for the cubic box3 + 3 blx2 + 3 b2x + b3 = 0.
Deduce the same result from V, ~ 35.
Ex. 8. By aid of ~ 35 compute the value of (a - /S)2(a- r)2(_ - 7)2
for the cubic x3 + x2 + x + 1 = 0. What relation has this symmetric
function to the discriminant of the cubic? How many values does the
function (a - /) (a -y) (/3 - y) assume when the roots are interchanged?
Why is this function not symmetric?
Ex. 9. Show that for the quartic
x4 ai -- 2 + a a 4 = 0,
(ala(t2 + a4)(ala + a2a4) ( ala4 + a2a3) = as2 + aL2a4 - 4 a2a4.
Ex. 10. Show that for this quartic
(a/ yS ( + 7a ) ( + + + a-) + ( + )( ) (d/3 + aa)(ya + 3- )
= a1a3 - 4 a4.
* Ex. 11. Form the cubic equation having for its roots
a/3 + ya, a(y + p/, p3y + aO.
Ex. 12. Show how the general quartic may be solved with the aid
of the roots of the cubic in Ex. 11 and the relation (a/Iy = a4.
Ex. 13. How many different values will the function a/3 + 7y assume,
as the roots are interchanged in every possible way?
* Ex. 14. Find the equation whose roots are
p- 2 + 5, p1 = V- 5, p2 = V/2 + W25,
p3 =- 2 + / 5, p4 =- 2 + / 5, p5 =- 2 + 2/5.
Let the required equation be
x6 + a1ix + a2x4 + a3a3 + a4X2 + a5X + a0 = 0.