The biquadratic equation,two of whose roots a | vedclass

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

Question

(A) Since the coefficients of the biquadratic equation are assumed to be rational,the irrational and complex roots must occur in conjugate pairs. Thus

Vedclass · Accepted Answer

$x^4-4 x^3+5 x^2-2 x-2=0$

Vedclass · Answer

(A) Since the coefficients of the biquadratic equation are assumed to be rational,the irrational and complex roots must occur in conjugate pairs. Thus,the roots are $1+i, 1-i, 1-\sqrt{2}, 1+\sqrt{2}$.For the roots $1+i$ and $1-i$:Sum $= (1+i) + (1-i) = 2$Product $= (1+i)(1-i) = 1^2 - i^2 = 1+1 = 2$The quadratic factor is $x^2 - 2x + 2 = 0$.For the roots $1-\sqrt{2}$ and $1+\sqrt{2}$:Sum $= (1-\sqrt{2}) + (1+\sqrt{2}) = 2$Product $= (1-\sqrt{2})(1+\sqrt{2}) = 1^2 - (\sqrt{2})^2 = 1-2 = -1$The quadratic factor is $x^2 - 2x - 1 = 0$.The biquadratic equation is the product of these two factors:$(x^2-2x+2)(x^2-2x-1) = 0$$x^2(x^2-2x-1) - 2x(x^2-2x-1) + 2(x^2-2x-1) = 0$$x^4 - 2x^3 - x^2 - 2x^3 + 4x^2 + 2x + 2x^2 - 4x - 2 = 0$$x^4 - 4x^3 + 5x^2 - 2x - 2 = 0$.

The biquadratic equation,two of whose roots are $1+i$ and $1-\sqrt{2}$,is

Similar Questions

If $\frac{1+\sqrt{3}i}{2}$ is a root of the equation $x^4-x^2+x-1=0$,then its real roots are:

Sodium metal reacts with $Al_2O_3$ at high temperature to give a sodium compound $X$. Compound $X$ reacts with $CO_2$ in water to give $Y$. What is $Y$?

$A$ bucket filled with water is tied to a rope of length $0.5 ~m$ and is rotated in a circular path in a vertical plane. The minimum velocity it should have at the lowest point of the circle so that the water does not spill is,$(g = 10 ~m/s^2)$.

In a galvanometer,$5 \%$ of the total current in the circuit passes through it. If the resistance of the galvanometer is $G$,the shunt resistance $S$ connected to the galvanometer is

The diagram below shows a hydraulic lift. $A$ force is applied at side $1$ and an output force is generated at side $2$. Which of the following is true?

Vedclass Test Series

Exam Paper Generator

Online Exam Module