Descartes' Rule of Signs

Learning Objectives State Descartes' Rule of Signs for both positive and negative real roots. Accurately count the number of sign variations in a polynomial P(x) and in P(-x). Apply the rule to determine the possible number of positive real roots for a given polynomial. Apply the rule to determine the possible number of negative real roots for a given polynomial. Construct a table summarizing all possible combinations of positive, negative, and imaginary roots for a polynomial. Explain how the total number of roots relates to the degree of the polynomial, in accordance with the Fundamental Theorem of Algebra. Ever wondered if you could predict the types of solutions a complex equation has without actually solving it? 🕵️‍♂️ Let's learn a clever shortcut from the 17t...

TermDefinitionExample Polynomial in Standard FormA polynomial where the terms are written in descending order of their exponents. The general form is P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0.P(x) = 5x^4 - 2x^3 + x - 9 is in standard form. P(x) = 3x - 4x^2 + 1 is not. Sign VariationAn instance where the signs of two consecutive non-zero coefficients in a standard-form polynomial are different.In P(x) = +3x^3 - 7x^2 - 2x + 1, the signs are (+, -, -, +). There are two sign variations: from +3 to -7, and from -2 to +1. Root (or Zero)A value 'c' for the variable 'x' such that the polynomial evaluates to zero, i.e., P(c) = 0. Roots are the solutions to the equation P(x) = 0.For P(x) = x^2 - 4, the roots are x = 2 and x = -2 because P(2)=0 and P(-2)=0. Real RootA...

Rule for Positive Real Roots Let P(x) be a polynomial with real coefficients. The number of positive real roots of P(x) = 0 is either equal to the number of sign variations in the coefficients of P(x), or is less than this number by an even integer (2, 4, 6, ...). Use this rule on the original polynomial P(x) as written. Count the sign changes between consecutive non-zero coefficients to find the maximum possible number of positive real roots. Rule for Negative Real Roots Let P(x) be a polynomial with real coefficients. The number of negative real roots of P(x) = 0 is either equal to the number of sign variations in the coefficients of P(-x), or is less than this number by an even integer (2, 4, 6, ...). First, you must find P(-x) by replacing every 'x' in the orig...