Let alpha and beta be two distinct roots of t | vedclass

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

Question

(A) Let the three roots of the equation $x^3 + 3x^2 - 1 = 0$ be $\alpha, \beta,$ and $\gamma$.From the properties of roots,the product of the roots is

Vedclass · Accepted Answer

$x^3 - 3x - 1 = 0$

Vedclass · Answer

(A) Let the three roots of the equation $x^3 + 3x^2 - 1 = 0$ be $\alpha, \beta,$ and $\gamma$.From the properties of roots,the product of the roots is $\alpha \beta \gamma = -(\frac{-1}{1}) = 1$.Thus,$\alpha \beta = \frac{1}{\gamma}$.Since $\gamma$ is a root of the original equation,it satisfies $\gamma^3 + 3\gamma^2 - 1 = 0$.Let $x = \alpha \beta = \frac{1}{\gamma}$,which implies $\gamma = \frac{1}{x}$.Substituting $\gamma = \frac{1}{x}$ into the original equation:$(\frac{1}{x})^3 + 3(\frac{1}{x})^2 - 1 = 0$$\frac{1}{x^3} + \frac{3}{x^2} - 1 = 0$Multiplying by $x^3$,we get $1 + 3x - x^3 = 0$,which simplifies to $x^3 - 3x - 1 = 0$.

Let $\alpha$ and $\beta$ be two distinct roots of the equation $x^3 + 3x^2 - 1 = 0$. The equation which has $(\alpha \beta)$ as its root is equal to

Similar Questions

If the sum of two roots of the equation $x^4-2x^3+x^2+4x-6=0$ is zero,then the sum of the squares of the other two roots is

If the sum of two roots of the equation $x^4+2x^3-7x^2-8x+12=0$ is zero,then the sum of the squares of the other two roots is

The minimum value of the sum of the squares of the roots of $x^{2}+(3-a)x+1=2a$ is:

Suppose the quadratic polynomial $p(x)=ax^2+bx+c$ has positive coefficients $a, b, c$ such that $b-a=c-b$. If $p(x)=0$ has integer roots $\alpha$ and $\beta$,then what could be the possible value of $\alpha+\beta+\alpha\beta$ if $0 \leq \alpha+\beta+\alpha\beta \leq 8$?

If $\alpha$ and $\beta$ are the roots of the equation $x^2 - 2x + 3 = 0$,then the equation whose roots are $\frac{1}{\alpha^2}$ and $\frac{1}{\beta^2}$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module