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According to Descartes' Rule of Signs, can the polynomial function have exactly $1$ positive real zero, including any repeated zeros? Choose your answer based on the rule only. $\newline$ $h(x) = x^4 + x^3 - 8x^2 + x + 9$ $\newline$ Choices: $\newline$ (A)yes $\newline$ (B)no

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Descartes' Rule of Signs

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Q. According to Descartes' Rule of Signs, can the polynomial function have exactly

1

positive real zero, including any repeated zeros? Choose your answer based on the rule only.

\newline

h(x) = x^4 + x^3 - 8x^2 + x + 9

\newline

Choices:

\newline

(A)yes

\newline

(B)no

Count Sign Changes: Count the number of sign changes in the coefficients of $h(x) = x^4 + x^3 - 8x^2 + x + 9$ . Coefficients: $1, 1, -8, 1, 9$ . Sign changes: $1$ to $-8$ , and $-8$ to $1$ . That's $2$ sign changes.
Descartes' Rule of Signs: According to Descartes' Rule of Signs, the number of positive real zeros is equal to the number of sign changes or less by an even number. $\newline$ So, $h(x)$ could have $2$ or $0$ positive real zeros.
Number of Positive Zeros: Since $h(x)$ can have $2$ or $0$ positive real zeros, it cannot have exactly $1$ positive real zero.

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