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

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

Vedclass · Accepted Answer

$pq-r$

Vedclass · Answer

(D) Given the cubic equation $x^3+px^2+qx+r=0$,the roots are $\alpha, \beta, \gamma$. By Vieta's formulas: $\alpha+\beta+\gamma = -p$ $\alpha\beta+\beta\gamma+\gamma\alpha = q$ $\alpha\beta\gamma = -r$ We know that $\alpha+\beta = -p-\gamma$,$\beta+\gamma = -p-\alpha$,and $\gamma+\alpha = -p-\beta$. Therefore,$(\alpha+\beta)(\beta+\gamma)(\gamma+\alpha) = (-p-\gamma)(-p-\alpha)(-p-\beta) = -(p+\gamma)(p+\alpha)(p+\beta)$. Let $f(x) = (x-\alpha)(x-\beta)(x-\gamma) = x^3+px^2+qx+r$. Then $f(-p) = (-p-\alpha)(-p-\beta)(-p-\gamma) = (-p)^3+p(-p)^2+q(-p)+r = -p^3+p^3-pq+r = r-pq$. Since $f(-p) = -(p+\alpha)(p+\beta)(p+\gamma)$,we have $-(p+\alpha)(p+\beta)(p+\gamma) = r-pq$. Thus,$(\alpha+\beta)(\beta+\gamma)(\gamma+\alpha) = pq-r$.

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

Similar Questions

For the equation $\frac{1}{x + a} - \frac{1}{x + b} = \frac{1}{x + c}$,if the product of the roots is zero,what is the sum of the roots?

If $\alpha$ and $\beta$ are the roots of the equation $ax^2 + bx + c = 0$,then what are the roots of the equation $cx^2 + bx + a = 0$?

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+4x+1=0$,then $(\alpha+\beta)^{-1}+(\beta+\gamma)^{-1}+(\gamma+\alpha)^{-1}$ is equal to

If $\alpha$ and $\beta$ are the roots of the equation $ax^2 + bx + c = 0$,then what is the value of $\frac{1}{(a\alpha + b)^2} + \frac{1}{(a\beta + b)^2}$?

If $\alpha$ and $\beta$ are the roots of $x^2+7x+3=0$ and $\frac{2\alpha}{3-4\alpha}, \frac{2\beta}{3-4\beta}$ are the roots of $ax^2+bx+c=0$ and $GCD(a, b, c) = 1$,then $a+b+c=$

Vedclass Test Series

Exam Paper Generator

Online Exam Module