Polynomials zzz (1 Viewer)

[JustRandomThings]

Hey every1 thnx in advance!

Denote a-alpha b-beta c-gamma

a,b and c are roots of the polynomial x^3-2x^2-5x-1=0. Find the equation with the roots 1/√a , 1/√b and 1/√c (1 on root alpha, 1 on root beta, 1 on root Gamma)

 

[pikachu975]

Hey every1 thnx in advance!

Denote a-alpha b-beta c-gamma

a,b and c are roots of the polynomial x^3-2x^2-5x-1=0. Find the equation with the roots 1/√a , 1/√b and 1/√c (1 on root alpha, 1 on root beta, 1 on root Gamma)

x=a,b,c
y= 1/sqrt(a) etc
y = 1/sqrt(x)
y^2 = 1/x
x = 1/y^2

Sub into the equation
(1/y^2)^3 - 2(1/y^2)^2 - 5(1/y^2) - 1 = 0
1/y^6 - 2/y^4 - 5/y^2 - 1 = 0
1 - 2y^2 - 5y^4 - y^6 = 0

Therefore
x^6 + 5x^4 + 2x^2 - 1 = 0

 

[JustRandomThings]

x=a,b,c
y= 1/sqrt(a) etc
y = 1/sqrt(x)
y^2 = 1/x
x = 1/y^2

Sub into the equation
(1/y^2)^3 - 2(1/y^2)^2 - 5(1/y^2) - 1 = 0
1/y^6 - 2/y^4 - 5/y^2 - 1 = 0
1 - 2y^2 - 5y^4 - y^6 = 0

Therefore
x^6 + 5x^4 + 2x^2 - 1 = 0

Hey can I ask how you obtained y=1/sqrt(x)

 

[fluffchuck]

If you have roots x=a, x=b, x=c and you want 1/sqrt(a), 1/sqrt(b) and 1/sqrt(c), subbing x=a, x=b and x=c into y=1/sqrt(x) will yield your desired roots

 

[JustRandomThings]

If you have roots x=a, x=b, x=c and you want 1/sqrt(a), 1/sqrt(b) and 1/sqrt(c), subbing x=a, x=b and x=c into y=1/sqrt(x) will yield your desired roots

Ahh I see now thanks Fluff and Pikachoooo!

 

[pikachu975]

Ahh I see now thanks Fluff and Pikachoooo!

Learnt this method in 4u but I guess you could probably use it in whichever maths this is.