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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. $\newline$ $f(x) = x^4 + 5x^3 + 2x^2 - 4x - 5$ $\newline$ 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

f(x) = x^4 + 5x^3 + 2x^2 - 4x - 5

\newline

Choices:

\newline

(A)yes

\newline

(B)no

Count Sign Changes: Count the number of sign changes in the coefficients of $f(x) = x^4 + 5x^3 + 2x^2 - 4x - 5$ . Coefficients: $1, 5, 2, -4, -5$ . Sign changes: $1$ (from $2$ to $-4$ ).
Descartes' Rule: 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. $\newline$ Since we have $1$ sign change, $f(x)$ can have either $1$ or $0$ positive real zeros.
Determine Positive Zero: Determine if $f(x)$ can have exactly $1$ positive real zero. $\newline$ Since the possible number of positive real zeros is $1$ or $0$ , it is possible for $f(x)$ to have exactly $1$ positive real zero.

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