Question

a) Find the nature of the roots of the polynomial 2x5 + x3 +5x +1.​

Answer 1

1 negative real roots and 4 imaginary roots.

Step-by-step explanation:

Given polynomial,

$2x^5+x^3+5x+1$

Assume $f(x)=2x^5+x^3+5x+1$      ...... (1)

Since in $f(x)$, the number of changes in sign = 0 ,

So, by the Descartes rule of sign,

Number of positive real roots = 0,

Now, substitute -x for x in equation (1),

$f(-x)=2(-x)^5+(-x)^3+5(-x)+1$

     $=-2x^5-x^3-5x+1$

In $f(-x)$, the sign changes from negative ( $-2x^5$ ) to negative ( $-x^3$ ), changes from negative ( $x^3$ ) to negative ( -5x ) then, finally it changes from negative (-5x) to positive (1).

Thus, in $f(-x)$,the number of changes in sign = 1,

So, by the Descartes sign rule,

Number of negative real roots = 1

Also, the degree of f(x) is 5.

That is, the total number of roots = 5,

Remaining roots = 5 - 1 = 4

Hence, imaginary roots = 4

#Learn more:

If two zeroes of the polynomial x3+3x2-5x-15 are root 5and -root 5 then find its third zero

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