Question Number 102883 by bobhans last updated on 11/Jul/20

If x^3 +ax^2 +bx+c = 0 has the roots are   α^�  β and γ . find the value of  αβ^2 +βγ^2 +γα^2  in terms a,b and c

Answered by bemath last updated on 11/Jul/20

α+β+γ = −a  αβ+αγ+βγ= b  αβγ = −c   ⇒(α+β+γ)(αβ+αγ+βγ) = −ab  ⇒α^2 β++α^2 γ+αβγ+αβ^2 +αβγ+β^2 γ+αβγ+αγ^2 +βγ^2 =−ab  αβ^2 +βγ^2 +γα^2 +α^2 β  +β^2 γ+αγ^2 +2αβγ = −ab  αβ^2 +βγ^2 +γα^2  =  −ab−2(−c)−β^2 γ−αγ^2 −α^2 β  = −ab+2c−(β^2 γ+αγ^2 +α^2 β)