Question

For the equation $p{x}^4+q{x}^3+r{x}^2+sx+t=0;p>0$, all the roots are positive real numbers, then which of the following is/are true?

• A. $t>0$
• B. $q<0$
• C. $s<0$
• D. $r>0$

Answer 1

Answer: (A), (B), (C), (D)

$r>0$

Given that $p{x}^4+q{x}^3+r{x}^2+sx+t=0;p>0$ and all the roots are positive real numbers.
The degree of the equation is $4$, therefore the no. of roots are also $4$.

The sum of the roots taken one at a time is given by $\frac{-q}{p}$
Since the roots are all positive, the sum of the roots taken one at a time will also be positive.
Since it is given that $p>0$, we can say that $q<0$

The sum of the roots taken two at a time is given by $\frac{r}{p}$
Since the roots are all positive, the sum of the roots taken two at a time will also be positive.
Since it is given that $p>0$, we can say that $r>0$

The sum of the roots taken three at a time is given by $\frac{-s}{p}$
Since the roots are all positive, the sum of the roots taken three at a time will also be positive.
Since it is given that $p>0$, we can say that $s<0$

The product of all roots is given by $\frac{t}{p}$
Since the roots are all positive, the product will also be positive.
Since it is given that $p>0$, we can say that $t>0$