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

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roshan-rox | (Level 1) Valedictorian

Posted June 25, 2013 at 3:21 AM via web

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

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degeneratecircle | High School Teacher | (Level 2) Associate Educator

Posted June 25, 2013 at 4:40 AM (Answer #1)

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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

`r_1^2+r_1r_2+r_1r_3=4r_1`

`r_1r_2+r_2^2+r_2r_3=4r_2`

`r_1r_3+r_2r_3+r_3^2=4r_3.`

Add these to get

`r_1^2+r_2^2+r_3^2+2(r_1r_2+r_1r_3+r_2r_3)=4(r_1+r_2+r_3),`

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

`r_1^2+r_2^2+r_3^2=4.`

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aruv | High School Teacher | (Level 2) Valedictorian

Posted June 25, 2013 at 5:12 AM (Answer #3)

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`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+gamma=-4`

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

`alpha beta gamma=-1`

using following identity

`A^2+B^2+C^2=(A+B+C)^2-2(AB+BC+CA)`

Thus

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

`=(-4)^2-2(6)`

`=16-12`

`=4`

Thus  our answer be

`alpha^2+beta^2+gamma^2=4`

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