If alpha and beta are two distinct negative r | vedclass

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

Question

(D) Given equation: $x^5-5x^3+5x^2-1=0$. By inspection,$x=1$ is a root. Factoring the polynomial: $(x-1)(x^4+x^3-4x^2+x+1)=0$. Further factoring: $(x-

Vedclass · Accepted Answer

$x^4-3x^2+1=0$

Vedclass · Answer

(D) Given equation: $x^5-5x^3+5x^2-1=0$. By inspection,$x=1$ is a root. Factoring the polynomial: $(x-1)(x^4+x^3-4x^2+x+1)=0$. Further factoring: $(x-1)^2(x^3+2x^2-2x-1)=0$. $(x-1)^3(x^2+3x+1)=0$. The roots are $x=1$ (triple root) and $x=\frac{-3 \pm \sqrt{5}}{2}$. The two distinct negative roots are $\alpha = \frac{-3+\sqrt{5}}{2}$ and $\beta = \frac{-3-\sqrt{5}}{2}$. We need an equation with roots $\sqrt{-\alpha}$ and $\sqrt{-\beta}$. Let $y = \sqrt{-\alpha} \Rightarrow y^2 = -\alpha$. Since $\alpha$ and $\beta$ are roots of $x^2+3x+1=0$,we have $\alpha+\beta = -3$ and $\alpha\beta = 1$. Let the new roots be $y_1 = \sqrt{-\alpha}$ and $y_2 = \sqrt{-\beta}$. The equation is $(y^2+\alpha)(y^2+\beta) = 0$. $y^4 + (\alpha+\beta)y^2 + \alpha\beta = 0$. Substituting the values: $y^4 - 3y^2 + 1 = 0$.

If $\alpha$ and $\beta$ are two distinct negative roots of $x^5-5x^3+5x^2-1=0$,then the equation of least degree with integer coefficients having $\sqrt{-\alpha}$ and $\sqrt{-\beta}$ as its roots is

Similar Questions

Let $(a-3)x^2+12x+(a+6)>0, \forall x \in R$ and $a \in (\ell, \infty)$. If $\alpha$ is the least positive integral value of $a$,then the roots of $(\alpha-3)x^2+12x+(\ell+2)=0$ are

If the quadratic equation $(a + b)x^2 + 2bx + 1 = 0$ has real and distinct roots,then a point having coordinates $(b, a)$:

Let $x$ and $y$ be two positive real numbers such that $x+y=1$. Then,the minimum value of $\frac{1}{x}+\frac{1}{y}$ is

Given the equation $4x^2 + 4(a - 1)x + (1 - 2a) = 0$ has roots $\sin \theta$ and $\cos \theta$ $(0 < \theta < \frac{\pi}{2})$,then the maximum value of $(a + \sin \theta)$ is-

The equation of lowest degree with rational coefficients having roots $\sqrt{3}+\sqrt{2} i$ and $\sqrt{3}-\sqrt{2}$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module