Find the cubic equation whose roots are the cubes of the roots of x^3+ax^2+bx+c=0,a,b,cinRR?
Find the cubic equation whose roots are the cubes of the roots of
x^3+ax^2+bx+c=0 ,a,b,cinRR
Find the cubic equation whose roots are the cubes of the roots of
1 Answer
Explanation:
Suppose the roots of the original cubic are
Then:
x^3+ax^2+bx+c = (x-alpha)(x-beta)(x-gamma)
color(white)(x^3+ax^2+bx+c) = x^3-(alpha+beta+gamma)x^2+(alphabeta+betagamma+gammaalpha)x-alphabetagamma
So we have:
{ (alpha+beta+gamma = -a), (alphabeta+betagamma+gammaalpha = b), (alphabetagamma = -c) :}
The cubic we are looking for is:
(x-alpha^3)(x-beta^3)(x-gamma^3)
= x^3-(alpha^3+beta^3+gamma^3)x^2+(alpha^3beta^3+beta^3gamma^3+gamma^3alpha^3)x-alpha^3beta^3gamma^3
So the problem essentially boils down to expressing each of the symmetric polynomials in
For example:
(alpha+beta+gamma)^3
=alpha^3+beta^3+gamma^3+3(alpha^2beta+beta^2gamma+gamma^2alpha+alphabeta^2+betagamma^2+gammaalpha^2)+6alphabetagamma
(alpha+beta+gamma)(alphabeta+betagamma+gammaalpha)
=alpha^2beta+beta^2gamma+gamma^2alpha+alphabeta^2+betagamma^2+gammaalpha^2+3alphabetagamma
So:
alpha^3+beta^3+gamma^3
=(alpha+beta+gamma)^3-3(alpha+beta+gamma)(alphabeta+betagamma+gammaalpha)+3alphabetagamma
=-a^3+3ab-3c
We also find:
(alphabeta+betagamma+gammaalpha)^3
= alpha^3beta^3+beta^3gamma^3+gamma^3alpha^3+3(alpha^3beta^2gamma+beta^3gamma^2alpha+gamma^3alpha^2beta+alpha^3betagamma^2+beta^3gammaalpha^2+gamma^3alphabeta^2)+6alpha^2beta^2gamma^2
(alpha+beta+gamma)(alphabeta+betagamma+gammaalpha)alphabetagamma
=alpha^3beta^2gamma+beta^3gamma^2alpha+gamma^3alpha^2beta+alpha^3betagamma^2+beta^3gammaalpha^2+gamma^3alphabeta^2+3alpha^2beta^2gamma^2
So:
alpha^3beta^3+beta^3gamma^3+gamma^3alpha^3
=(alphabeta+betagamma+gammaalpha)^3-3(alpha+beta+gamma)(alphabeta+betagamma+gammaalpha)alphabetagamma+3(alphabetagamma)^2
=b^3-3abc+3c^2
Finally:
alpha^3beta^3gamma^3 = (alphabetagamma)^3 = -c^3
Hence the required cubic equation is:
x^3+(a^3-3ab+3c)x^2+(b^3-3abc+3c^2)x+c^3 = 0
Footnote
If you would like to see a more advanced application of symmetric polynomials, you may like to take a look at this one: https://socratic.org/s/aCWXbG2b
