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According to Descartes' Rule of Signs, can the polynomial function have exactly $2$ positive real zeros, including any repeated zeros? Choose your answer based on the rule only. $\newline$ $g(x) = x^4 - x^3 - 8x^2 - 6x + 8$ $\newline$ Choices: $\newline$ (A)yes $\newline$ (B)no

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Descartes' Rule of Signs

Full solution

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

2

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

\newline

g(x) = x^4 - x^3 - 8x^2 - 6x + 8

\newline

Choices:

\newline

(A)yes

\newline

(B)no

Count Sign Changes: Count the number of sign changes in the coefficients of $g(x) = x^4 - x^3 - 8x^2 - 6x + 8$ . Coefficients: $1, -1, -8, -6, 8$ . Sign changes: $1$ (from $1$ to $-1$ ), $2$ (from $-6$ to $8$ ).
Apply 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. $\newline$ We have $2$ sign changes, so $g(x)$ can have $2$ or $0$ positive real zeros.
Determine Real Zeros: Since $g(x)$ can have $2$ positive real zeros, the answer to the question is yes.

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