math problem

Question

The Depressed Cubic The most general cubic (thirddegree)
equation with rational coefficients can be written
as
x^3+ax^2+bx+c=0

(a) Show that if we replace x by X a/3 and simplify, we
end up with an equation that doesn’t have an X2 term,
that is, an equation of the form
x^3+px+q=0

 

This is called a depressed cubic, because we have
“depressed” the quadratic term.
(b) Use the procedure described in part (a) to depress the
equation  x^3+6x^2+9x+4=0

Answer 1

(a). Replace x by X - a/3:

 

(X-a/3)³ +a(X-a/3)² +b(X-a/3)+c = 0

 

X³-aX²+(a²/3)X-(a³/27)+a[X² - (2a/3)X +a²/9]+bX-ab/3+c = 0

 

X³ +(-a²/3+b)X +(-a³/27 + a³/9 - ab/3 + c) = 0

 

(b).Replace x by X - a/3 = X - 6/3 = X - 2:

 

(X-2)³ +6(X-2)²+9(X-2)+4 = 0

 

X³ - 6X² + 12X - 8 + 6(X² - 4X + 4) + 9X - 18 + 4 = 0

 

X³ - 3X + 2 = 0