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According to Descartes' Rule of Signs, can the polynomial function have exactly 55 positive real zeros, including any repeated zeros? Choose your answer based on the rule only.\newlineg(x)=x5−4x3−4x2+9g(x) = x^5 - 4x^3 - 4x^2 + 9\newlineChoices:\newline(A)yes\newline(B)no

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Full solution

Q. According to Descartes' Rule of Signs, can the polynomial function have exactly 55 positive real zeros, including any repeated zeros? Choose your answer based on the rule only.\newlineg(x)=x5−4x3−4x2+9g(x) = x^5 - 4x^3 - 4x^2 + 9\newlineChoices:\newline(A)yes\newline(B)no
  1. Count Sign Changes: Count the number of sign changes in the coefficients of g(x)=x5−4x3−4x2+9g(x) = x^5 - 4x^3 - 4x^2 + 9. Coefficients: 1,0,−4,−4,0,91, 0, -4, -4, 0, 9. Sign changes: 11 (from −4-4 to 99).
  2. 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.\newlineSo, g(x)g(x) can have 11 or 1−2=−11-2= -1 positive real zeros, but since we can't have negative zeros, we only consider 11.
  3. Number of Positive Zeros: Since g(x)g(x) can have only 11 positive real zero, it cannot have exactly 55 positive real zeros.

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