Question

A polynomial $f\left(x\right)-a_4{x}^4+a_3{x}^3+a_2{x}^2+a_{1}x-a_0$ with all coefficients positive has

• A. no real roots
• B. no negative real root
• C. odd number of real roots
• D. at least one positive and one negative real root

Answer 1

The correct option is D at least one positive and one negative real root

$f\left(x\right)=a_4{x}^4+a_3{x}^3+a_2{x}^2+a_{1}x-a_0$
$a_0,a_1,a_2,a_3,a_4>0$
$f\left(x\right)$ has one change of sign
So $f\left(x\right)$ will have maximum one +ve real root
$f(-x)=a_4{x}^4-a_3{x}^3+a_2{x}^2+a_{1}x-a_0$
$f(-x)$ has $3$ changes of sign
So, $f\left(x\right)$ has at most $3$ -ve real root
$\therefore f\left(x\right)$ has atleast one positive and one negative real root.