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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)=x5−9x4−4x2+2x+8g(x) = x^5 - 9x^4 - 4x^2 + 2x + 8\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)=x5−9x4−4x2+2x+8g(x) = x^5 - 9x^4 - 4x^2 + 2x + 8\newlineChoices:\newline(A)yes\newline(B)no
  1. Count Sign Changes: Count the number of sign changes in the coefficients of g(x)=x5−9x4−4x2+2x+8g(x) = x^5 - 9x^4 - 4x^2 + 2x + 8. Coefficients: 1,−9,0,−4,2,81, -9, 0, -4, 2, 8. Sign changes: 11 to −9-9 (one change), −4-4 to 22 (second change), 22 to 88 (no change). Total sign changes: 22.
  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 than that by an even number.\newlineSo, g(x)g(x) can have 22 or 00 positive real zeros.
  3. Check Positive Zeros: Check if g(x)g(x) can have exactly 55 positive real zeros.\newlineSince g(x)g(x) can have at most 22 positive real zeros, it cannot have exactly 55 positive real zeros.

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