Question 7 of 10 , Step 1 of 1
Correct

Use Descartes' Rule of Signs to determine the possible number of positive and negative real zeros. Be sure to include all possibilities.
\[
g(x)=9 x^{12}+3 x^{10}+6 x^{8}+7 x^{6}+7
\]

Answer
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Separate multiple answers with commas.
Number of Positive Real Zeros:
Number of Negative Real Zeros:

Step 1 ：Use Descartes' Rule of Signs to determine the possible number of positive and negative real zeros. The rule states that the number of positive real zeros of a polynomial is equal to the number of sign changes between consecutive non-zero coefficients, or less than that by an even number. The number of negative real zeros is found by applying the rule to the polynomial obtained by replacing x by -x.

Step 2 ：In this case, the polynomial is \(g(x)=9 x^{12}+3 x^{10}+6 x^{8}+7 x^{6}+7\). We can see that there are no sign changes in the coefficients, so there are no positive real zeros.

Step 3 ：To find the number of negative real zeros, we replace x by -x to get \(g(-x)=9 (-x)^{12}+3 (-x)^{10}+6 (-x)^{8}+7 (-x)^{6}+7\). Simplifying this gives \(g(-x)=9 x^{12}+3 x^{10}+6 x^{8}+7 x^{6}+7\), which is the same as the original polynomial. Therefore, there are also no negative real zeros.

Step 4 ：Final Answer: The number of positive real zeros is \(\boxed{0}\) and the number of negative real zeros is also \(\boxed{0}\).