Possible positive real zero 0 or 2
Possible negative real zeros: 1
Total complex (including imaginary) zeros: 7
To determine the possible number of positive real zeros, negative real zeros, and imaginary zeros for the function g(x)=x^7 +4x^4 −10x+25, we can use Descartes' Rule of Signs and the Fundamental Theorem of Algebra.
Descartes' Rule of Signs
Number of Positive Real Zeros:
Count the sign changes in g(x) and g(−x):
g(x)=x^7 +4x^4 −10x+25
There are two sign changes in g(x) (from x^7 to 4x^4 and then from 4x^4 to −10x).
When we substitute −x into g(x) to find g(−x), there's only one sign change (from (−x)^7 to −10(−x)).
Therefore, the number of positive real zeros can be 2 or 0 (the actual number could be 2 or 0 but not any other even number).
Number of Negative Real Zeros:
The number of sign changes in g(−x) is the number of negative real zeros or a difference of an even number. We've already found 1 sign change, indicating there's exactly 1 negative real zero.
Using the Fundamental Theorem of Algebra
The total number of complex zeros (which includes imaginary zeros) for a polynomial function is given by the degree of the polynomial.
For g(x)=x^7 +4x^4 −10x+25:
The degree of the polynomial is 7.
Therefore, there are 7 complex zeros in total (including real and imaginary).