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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.\newlineh(x)=x4−x3−7x2+x−1h(x) = x^4 - x^3 - 7x^2 + x - 1\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.\newlineh(x)=x4−x3−7x2+x−1h(x) = x^4 - x^3 - 7x^2 + x - 1\newlineChoices:\newline(A) yes\newline(B) no
  1. Count Sign Changes: Count the number of sign changes in the coefficients of h(x)=x4−x3−7x2+6x−1h(x) = x^4 - x^3 - 7x^2 + 6x - 1. Coefficients: 1,−1,−7,6,−11, -1, -7, 6, -1. Sign changes: 11 to −1-1 (one change), −7-7 to 66 (second change), 66 to −1-1 (third change). Total sign changes: 33.
  2. Descartes' Rule of Signs: According to Descartes' Rule of Signs, the number of positive real zeros is either equal to the number of sign changes or less than that by an even number.\newlineWith 33 sign changes, the possible numbers of positive real zeros are 33 or 11.
  3. Check Possibility: Check if having exactly 44 positive real zeros is possible.\newlineSince the possible numbers of positive real zeros are 33 or 11, having exactly 44 positive real zeros is not possible.

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