What is the sum of the squares of the roots of the cubic equation `x^3 -4x^2 +6x+1`

Expert Answers
degeneratecircle eNotes educator| Certified Educator

Equating coefficients gives

`r_1+r_2+r_3=4`                     (1)

`r_1r_2+r_1r_3+r_2r_3=6`        (2)

Multiply equation (1) by `r_1,` then `r_2,` then `r_3.` This gives three new equations




Add these to get


and using our earlier equations (1) and (2), we can solve to get


aruv | Student

`x^3-4x^2+6x+1=0`  be given equation.

Let `alpha,beta , and gamma`  be the roots of above cubic equation.

By  relation between roots and coefficients ,we have


`alpha beta+beta gamma+gamma alpha=6`

`alpha beta gamma=-1`

using following identity



`alpha^2+beta^2+gamma^2=(alpha+beta+gamma)^2-2(alpha beta+beta gamma+gamma alpha)`




Thus  our answer be