Determine the Possible Number of Real Roots f(x)=11x^5+5x+25


Step 1
To find the possible number of positive roots, look at the signs on the coefficients and count the number of times the signs on the coefficients change from positive to negative or negative to positive.

Step 2
Since there are sign changes from the highest order term to the lowest, there are at most positive roots (Descartes\' Rule of Signs).
Positive Roots:

Step 3
To find the possible number of negative roots, replace with and repeat the sign comparison.

Step 4
Simplify each term.

Apply the product rule to .
Raise to the power of .
Multiply by .
Multiply by .

Step 5
Since there is sign change from the highest order term to the lowest, there is at most negative root (Descartes\' Rule of Signs).
Negative Roots:

Step 6
The possible number of positive roots is , and the possible number of negative roots is .
Positive Roots:
Negative Roots:


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