Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative
real zeros for the function.
-$$f ( x ) = 7 x ^ { 8 } + 3 x ^ { 6 } + 2 x ^ { 4 } + 5 x ^ { 2 } + 2$$
A)
$\begin{array}{c|c|c}
\text { Positive } & \text { Negative } & \text { Nonreal Complex } \
\hline 4 & 4 & 0
\end{array}$

B)
$\begin{array}{c|c|c}
\text { Positive } & \text { Negative } & \text { Nonreal Complex } \
\hline 4 & 0 & 4
\end{array}$

C)
$\begin{array}{c|c|c}
\text { Positive } & \text { Negative } & \text { Nonreal Complex } \
\hline 0 & 4 & 4
\end{array}$

D)
$\begin{array}{c|c|c}
\text { Positive } & \text { Negative } & \text { Nonreal Complex } \
\hline 0 & 0 & 8
\end{array}$

Question

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative
real zeros for the function.
-$$f ( x ) = 7 x ^ { 8 } + 3 x ^ { 6 } + 2 x ^ { 4 } + 5 x ^ { 2 } + 2$$
A)
$\begin{array}{c|c|c}
\text { Positive } & \text { Negative } & \text { Nonreal Complex } \
\hline 4 & 4 & 0
\end{array}$

B)
$\begin{array}{c|c|c}
\text { Positive } & \text { Negative } & \text { Nonreal Complex } \
\hline 4 & 0 & 4
\end{array}$

C)
$\begin{array}{c|c|c}
\text { Positive } & \text { Negative } & \text { Nonreal Complex } \
\hline 0 & 4 & 4
\end{array}$

D)
$\begin{array}{c|c|c}
\text { Positive } & \text { Negative } & \text { Nonreal Complex } \
\hline 0 & 0 & 8
\end{array}$

Quizplus · Accepted Answer

Descartes' Rule of Signs indicates that the number of positive real zeros is equal to the number of sign changes in $f(x)$ or less than that by an even number. Since $f(x) = 7x^8 + 3x^6 + 2x^4 + 5x^2 + 2$ has no sign changes, there are 0 positive real zeros. For negative real zeros, substitute $x$ with $-x$ and count the sign changes. Since all terms in $f(x)$ are even powers of $x$, substituting $-x$ for $x$ results in no sign changes, indicating 0 negative real zeros. The remaining 8 zeros must be nonreal complex.

Use Descartes' Rule of Signs to Determine the Possible Number f(x)=7x8+3x6+2x4+5x2+2f ( x ) = 7 x ^ { 8 } + 3 x ^ { 6 } + 2 x ^ { 4 } + 5 x ^ { 2 } + 2f(x)=7x8+3x6+2x4+5x2+2

Use Descartes' Rule of Signs to Determine the Possible Number $f ( x ) = 7 x ^ { 8 } + 3 x ^ { 6 } + 2 x ^ { 4 } + 5 x ^ { 2 } + 2$