Question

Find the nature of the roots of the equation
$3x^4+12x^2+5x-4=0$

Answer 1

Give equation, $3x^4+12x^2+5x-4=0$, has one changes of signs. Hence there can be a maximum of 1 positive roots.

$f(-x)=3x^4+12x^2-5x-4=0$, which has one changes of signs. Hence the given equation has a maximum of 1 negative roots.

Now, as the equation is of $4^{th}$ degree, it must have at least $(4-1-1)=2$ imaginary roots.

Also since the degree of the equation is even and the constant term is negative, it must have atleast one positive real and one negative real roots.

$\therefore$ The given equation has one positive, one negative and two imaginary roots