Question

Select the correct statements for the equation $x^5+4x^4-3x^2+x-6=0$ ?

• A. The number of positive roots can be $3$
• B. The number of positive roots can be $4$
• C. The number of positive roots can be $1$
• D. The number of positive roots can be $2$

Answer 1

Answer: (A), (C)

The number of positive roots can be $1$

Let $f\left(x\right)=x^5+4x^4-3x^2+x-6$.
The number of sign changes in $f\left(x\right)$ is $3.$
Now, using Descartes rule of sign:
The maximum possible number of positive real roots = Number of sign changes in $f\left(x\right)$ when the terms are arranged in the canonical order i.e. $3.$

Now, the possible number of positive real roots will be a decrement by a multiple of 2 from the total number of sign changes in $f\left(x\right)$ when when the terms are arranged in the canonical order.
$\Rightarrow$ the possible number of positive real roots can be $1$ or $3$.