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According to Descartes' Rule of Signs, can the polynomial function have exactly 44 positive real zeros, including any repeated zeros? Choose your answer based on the rule only.\newlineg(x)=x7+5x6−8x5−4x4+9x3−5x−4g(x) = x^7 + 5x^6 - 8x^5 - 4x^4 + 9x^3 - 5x - 4\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 44 positive real zeros, including any repeated zeros? Choose your answer based on the rule only.\newlineg(x)=x7+5x6−8x5−4x4+9x3−5x−4g(x) = x^7 + 5x^6 - 8x^5 - 4x^4 + 9x^3 - 5x - 4\newlineChoices:\newline(A)yes\newline(B)no
  1. Count Sign Changes: Count the number of sign changes in the coefficients of g(x)=x7+5x6−8x5−4x4+9x3−5x−4g(x) = x^7 + 5x^6 - 8x^5 - 4x^4 + 9x^3 - 5x - 4. Coefficients: 1,5,−8,−4,9,−5,−41, 5, -8, -4, 9, -5, -4 Sign changes: 11 (from 55 to −8-8), 22 (from −4-4 to 99), 33 (from 99 to 1,5,−8,−4,9,−5,−41, 5, -8, -4, 9, -5, -400).
  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 33, 11, or 00 positive real zeros.
  3. Number of Positive Zeros: Since g(x)g(x) can have at most 33 positive real zeros and not 44, the answer is no.

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