Descartes' Rule of Signs

Algebra 2 · Axiom Academy

LESSON Descartes' Rule of Signs Bound the positive and negative real zeros of a polynomial just by counting the sign changes in its coefficients. Descartes' Rule of Signs reads the signs of a polynomial's coefficients, in order from the highest-degree term down to the constant, and turns them into a count of real zeros. The number of positive real zeros equals the number of sign changes between consecutive nonzero coefficients of f(x) , or fewer than that by an even number . The number of negative real zeros equals the number of sign changes in f(-x) , again or fewer by an even number . A sign change happens when two consecutive nonzero coefficients have opposite signs (+ to − or − to +). Zero coefficients are skipped — you compare the nearest nonzero neighbours. Take a concrete polynomial and read its coefficient signs in order, comparing each neighbouring pair: 3. Example: Positive Real Zeros Count the sign changes in f(x) = x^3 - 4x^2 - 3x + 18 , then watch the prediction confirmed where the graph meets the x -axis. Example: f(x) = x^3 - 4x^2 - 3x + 18 This one factors as f(x) = (x-3)^2(x+2) , so it has exactly 2 positive real zeros , both at x = 3 (a double root) — the upper bound of 2 is met, and x = -2 is the one negative zero. 4. Finding Negative Real Zeros For the negative zeros, substitute -x for x and count the sign changes in f(-x) . Odd-degree terms flip sign; even-degree terms keep theirs. Continuing f(x) = x^3 - 4x^2 - 3x + 18

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