Question

Find nature of root of polynomial 2x^5+x^3+5x+1

Answer 1

Let the given polynomial be $f(x)=2x^5+x^3+5x+1$ .

Since it is an odd degree polynomial ,hence it has at least one real roots.

If possible ,$f(x)$ have two real roots $a \ and \ b$ .

Then $f(a)=f(b)=0$ .

Again, since polynomial function are continuous and differential over $\R$ $($ In particular over any subset of $\R)$

Therefore $1)f(x)$ is continuous at $[a,b]$

$2) f(x)$ is differentiable at $(a,b)$

and $f(a)=f(b)=0$ .

Then by Rolle's Theorem, there exit at least one point $c\in(a,b)$ such that $f'(c)=0$ .

But $f'(c)=10x^4+3x^2+5 >0$ ,since $x^4\geq0 \ and \ x^2\geq 0$ for any real number $x$

Therefore,We get a contradiction.

Hence ,$f(x)$ have exactly one real root .

Also the number of imaginary roots$=5-1=4$ .

Since ,a polynomial of degree n over a field of complex number have exactly n roots.