Question

The equation $x^4-5x^3+k{x}^2-3x-1=0$ has

• A. $4$ real and distinct roots
• B. Two real and two imaginary roots
• C. $4$ imaginary roots
• D. Two repeated real roots and two imaginary roots

Answer 1

Answer: (A)

$4$ real and distinct roots

$\alpha ,\beta ,\gamma ,\delta$ $x^4-5x^3+k{x}^2-3x-1=-0$

$\alpha ,\beta$ $x^2-ax-1=0$

$\gamma \delta$ $x^2-bx+1=0$

$\alpha +\beta +\gamma +\delta =5$ $a+b=5$

$\alpha +\beta =+a$ $\alpha \beta +\beta \gamma +\gamma \delta +\delta \alpha =k$

$\gamma +\delta =b$ $1+1+\beta \gamma +\delta \alpha =k$

$\alpha \beta \gamma \delta =-1$

$\alpha \beta =-1$

$\gamma \delta =1$

Hence sthey are satisfying $\alpha \beta \gamma \delta =-1$

$4$ real & distinct roots.