Decartes Rule of Signs

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Rene Descartes came up with a rule of signs that can be used to find the number of possible positive and negative solutions to a polynomial equation. Positive Roots The rule states that if the terms of a single-variable polynomial with real coefficients are ordered by descending variable exponent, then the number of positive roots of the polynomial is either equal to the number of sign differences between consecutive nonzero coefficients, or less than it by a multiple of 2. Multiple roots of the same value are counted separately. Negative Roots As a corollary of the rule, the number of negative roots is the number of sign changes after negating the coefficients of odd-power terms (otherwise seen as substituting the negation of the variable for the variable itself), or fewer than it by a multiple of 2. {Wikipedia} To use type in an expression of any 6th degree or less polynomial into the input box, f(x). To clear click on the Reset Button or if the yellow fields do not work click on the reset button.
Colin Maclaurin (February 1698 – 14 June 1746) developed the MacLaurin Series; using this series one can extract the coefficients from f(x). The coefficients are a0 = f(0), a1 = f'(0), a2 = f''(0)/2!, a3 = f'''(0)/3!, a4 = f''''(0)/4!, etc. This is a topic that you will learn in Calculus, and it has been used to extra data from f(x) for the spreadsheet attached in the first row. The formulas for the second row provide the equivalent of replacing x with a -x in the function. This applet can be used by teachers in a demonstration mode in the classroom or teachers can have students load it on their own computers as a worksheet to be completed for a grade.