Finding root of equation

Question

This question was asked in one of the enterance test of mathematics in India which is For the equation $1+2x+x^{3}+4x^{5}=0$, which of the following is true?

(A) It does not possess any real root

(B) It possesses exactly one real root

(C) It possesses exactly two real roots

(D) It possesses exactly three real roots.

Answer 1

HINTS:

$1$. A polynomial with odd degree has at-least one real root.

$2$. A strictly monotone increasing function can have at-most one real root.

$3$. If a function's derivative is strictly positive then it is monotonically increasing.

Answer 2

Hint: Look at the polynomial's value in plus and minus infinity, and look at the derivative.

Answer 3

Hint: Apply Descartes' rule of signs.

Answer 4

A polynomial of odd degree always have a real root since if $p(x)$ is such (WLOG monic) polynomial then $$\lim_{x\to\infty}p(x)=\infty$$ and $$\lim_{x\to-\infty}p(x)=-\infty$$ and by IVT there is $x_{0}\in\mathbb{R}$ s.t $p(x_{0})=0$.

Now, the derivative in your case is $$20x^{4}+3x^{2}+2>0$$ hence there is only one root.