Question

If $\alpha \beta \gamma$ are the zeros of the polynomial $f(x)=ax^{3}+bx^{2}+cx+d$, then $\alpha^{2} + \beta^{2} + \gamma^{2}$=
(a) $\frac{b^{2}-ac}{a^{2}}$
(b) $\frac{b^{2}-2ac}{a}$
(c) $\frac{b^{2}+2ac}{b^{2}}$
(d) $\frac{b^{2}-2ac}{a^{2}}$

Answer 1

SOLUTION :

The correct option is (d) :   (b² - 2ca)/a² .

Given :  α, β, γ are the three Zeroes of the cubic  polynomial f(x) = ax³ + bx² + cx + d

On comparing with ax³ + bx² + cx + d,

a = a , b= b ,c = c , d = d

Sum of zeroes =−coefficient of x² / coefficient of x³

α + β + γ = −b/a ………………….(1)

Sum of the product  of its zeroes taken two at a time = coefficient of x / coefficient of x³

αβ + βγ + αγ = c/a ………….(2)

The value of : α² + β² + γ²  

= (α + β + γ)² - 2(αβ + βγ + αγ)

[By using the Identity : (a + b + c ) ² = a² + b² + c² + 2 (ab + bc + ac) ]

= (- b/a)² - 2(c/a)

= b²/a² - 2c/a

= (b² - 2ca)/a²

The value of α² + β² + γ² is  (b² - 2ca)/a² .

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Answer 2

option (d) is correct