Basic Technical Mathematics, 11th Edition

15.3 Rational and Irrational Roots

• Rational Root Theorem • Descartes’ Rule of Signs • Roots of a Polynomial Equation

The product $(x\hspace{0.17em}+\hspace{0.17em}2)(x\hspace{0.17em}-\hspace{0.17em}4)(x\hspace{0.17em}+\hspace{0.17em}3)$ equals $x^3\hspace{0.17em}+\hspace{0.17em}{x}^2\hspace{0.17em}-\hspace{0.17em}14x\hspace{0.17em}-\hspace{0.17em}24\hspace{0.17em}.\hspace{0.17em}$ Here, we find that the constant 24 is determined only by the 2, 4, and 3. These numbers represent the roots of the equation if the given function is set equal to zero. In fact, if we find all the integer roots of an equation with integer coefficients and represent the equation in the form

$$f(x)\hspace{0.17em}=\hspace{0.17em}(x\hspace{0.17em}-\hspace{0.17em}{r}_1)(x\hspace{0.17em}-\hspace{0.17em}{r}_2)\hspace{0.17em}\cdots \hspace{0.17em}(x\hspace{0.17em}-\hspace{0.17em}{r}_k)f_{k\hspace{0.17em}+\hspace{0.17em}1}(x)\hspace{0.17em}=\hspace{0.17em}0$$

where all the roots indicated are integers, the constant term of f(x) must have factors of $r_1\hspace{0.17em},\hspace{0.17em}{r}_2\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\dots \hspace{0.17em}\hspace{0.17em},\hspace{0.17em}{r}_k\hspace{0.17em}.\hspace{0.17em}$ This leads us to the theorem that states that

in a polynomial equation $f(x)\hspace{0.17em}=\hspace{0.17em}0\hspace{0.17em},\hspace{0.17em}$ if the coefficient of the highest ...