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
1. No real roots
2. No negative real root
3. Odd no. of real roots
4. At least one positive and one negative real root.

Answer 1

Correct Answer - Option 4 : At least one positive and one negative real root.

Using Routh array

X­4	a­4	a­2	-a­0
X­3	a­3	a­1
X­2	\(\frac{{{a_3}{a_2} - {a_4}{a_1}}}{{{a_3}}}\)	-a­0
X­1	\(\frac{{p{a_1} - {a_3}{a_0}}}{p}\)
X­0	-a­0

There is at least one sign change hence at least one positive root. Hence (d) is correct.