Question

If α, β, γ are the zeros of the polynomial f(x) = ax³ + bx² + cx + d, then α² + β² + γ² =

Options

• \[\frac{b^2 - ac}{a^2}\]
• \[\frac{b^2 - 2ac}{a}\]
• \[\frac{b^2 + 2ac}{b^2}\]
• \[\frac{b^2 - 2ac}{a^2}\]

Answer 1

We have to find the value of `alpha^2+beta^2+y^2`

Given `alpha,beta,y` be the zeros of the polynomial f(x) = ax³ + bx² + cx + d,

`alpha + ß + y= - (-text{coefficient of }x^2)/(text{coefficient of } x^3)`

`= (-b)/a`

`alphaß +betay+yalpha=  (text{coefficient of x})/(text{coefficient of } x^3)`

`= c/a`

Now

`alpha^2+beta^2+y^2=(alpha+beta+y)^2-2(alphabeta+betay+yalpha)`

`alpha^2+beta^2+y^2=((-6)/a)^2-2(c/a)`

`alpha^2+b^2+y^2= (b^2)/(a^2)-(2c)/a`

`alpha^2+beta^2+y^2=(b^2)/(a^2)- (2cxxa)/(axxa) `

`alpha^2+beta^2+y^2=(b^2)/(a^2)- (2ca)/a^2 `

`alpha^2+beta^2+y^2=(b^2)/(a^2)- (b^2-2ac)/a^2`

The value of `alpha^2+beta^2+y^2=( b^2-2ac)/a^2`

Hence, the correct choice is  `(d).`