If alpha, beta, gamma are the roots of the eq | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Given the cubic equation $x^3+px^2+qx+r=0$,the roots are $\alpha, \beta, \gamma$. From Vieta's formulas: $\alpha+\beta+\gamma = -p$ $\alpha\beta+\

Vedclass · Accepted Answer

$3pq-3r-p^3$

Vedclass · Answer

(C) Given the cubic equation $x^3+px^2+qx+r=0$,the roots are $\alpha, \beta, \gamma$. From Vieta's formulas: $\alpha+\beta+\gamma = -p$ $\alpha\beta+\beta\gamma+\gamma\alpha = q$ $\alpha\beta\gamma = -r$ We use the identity: $\alpha^3+\beta^3+\gamma^3-3\alpha\beta\gamma = (\alpha+\beta+\gamma)(\alpha^2+\beta^2+\gamma^2-\alpha\beta-\beta\gamma-\gamma\alpha)$ We know that $\alpha^2+\beta^2+\gamma^2 = (\alpha+\beta+\gamma)^2 - 2(\alpha\beta+\beta\gamma+\gamma\alpha) = (-p)^2 - 2q = p^2-2q$. Substituting these into the identity: $\alpha^3+\beta^3+\gamma^3 - 3(-r) = (-p)((p^2-2q) - q)$ $\alpha^3+\beta^3+\gamma^3 + 3r = -p(p^2-3q)$ $\alpha^3+\beta^3+\gamma^3 = -p^3+3pq-3r$ $\alpha^3+\beta^3+\gamma^3 = 3pq-3r-p^3$.

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+px^2+qx+r=0$,then $\alpha^3+\beta^3+\gamma^3=$

Similar Questions

If the product of the roots of the equation $x^2 - 3kx + 2e^{2\log k} - 1 = 0$ is $7$,then find the value of $k$.

If $\sin 2 \theta$ and $\cos 2 \theta$ are solutions of $x^2+bx-c=0$,then

If $a, b$ and $c$ are the roots of $x^3+qx+r=0$,then $(a-b)^2+(b-c)^2+(c-a)^2=$ (in $q$)

If the harmonic mean of the roots of the equation $\sqrt{2} x^2 - bx + (8 - 2\sqrt{5}) = 0$ is $4$,then the value of $b$ is

If $\alpha, \beta, \gamma$ are roots of the equation $x^3+a x^2+b x+c=0$,then $\alpha^{-1}+\beta^{-1}+\gamma^{-1} = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module