Question

Find the nature of the roots of the polynomial 2x^5+x^3+5x +1.​​

Answer 1

Answer:

Given polynomial,

2x^5+x^3+5x+12x

5

+x

3

+5x+1

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

5

+x

3

+5x+1 ...... (1)

Since in f(x)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)+1f(−x)=2(−x)

5

+(−x)

3

+5(−x)+1

=-2x^5-x^3-5x+1=−2x

5

−x

3

−5x+1

In f(-x)f(−x) , the sign changes from negative ( -2x^5−2x

5

) to negative ( -x^3−x

3

), changes from negative ( x^3x

3

) to negative ( -5x ) then, finally it changes from negative (-5x) to positive (1).

Thus, in f(-x)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