## College Algebra (6th Edition)

Descartes's Rule of Signs (page 384) Let $f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{2}x^{2}+a_{1}x+a_{0}$ be a polynomial with real coefficients. 1. The number of positive real zeros of $f$ is either $\mathrm{a}$. the same as the number of sign changes of $f(x)$ or $\mathrm{b}$. less than the number of sign changes of $f(x)$ by a positive even integer. If $f(x)$ has only one variation in sign, then $f$ has exactly one positive real zero. 2. The number of NEGATIVE real zeros of $f$ is either $\mathrm{a}$. the same as the number of sign changes of $f(-x)$ or $\mathrm{b}$. less than the number of sign changes of $f(-x)$ by a positive even integer. If $f(-x)$ has only one variation in sign, then $f$ has exactly one negative real zero. ----------------- So, Descartes's Rule generally gives ESTIMATES, and the MAXIMUM number of positive/negative real roots. It gives an exact number (one) only when the number of sign variations in f(x) or f(-x) is 1 The statement is false. Two ways of making it true is changing " ... gives the exact number ..." to "... gives the maximum number ..." or "gives an estimate "