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