Basic Technical Mathematics, 11th Edition

15.1 The Remainder and Factor Theorems; Synthetic Division

• Polynomial Function • Remainder Theorem • Factor Theorem • Synthetic Division

In solving higher-degree polynomial equations, the quadratic formula can be used for second-degree equations, and methods have been found for certain third- and fourth-degree equations. It can also be proven that polynomial equations of degree higher than 4 cannot in general be solved algebraically.

In this section, we present two theorems and a simplified method for algebraic division. These will help us in solving polynomial equations later in the chapter.

Any function of the form

$$f(x)\hspace{0.17em}=\hspace{0.17em}{a}_n{x}^n\hspace{0.17em}+\hspace{0.17em}{a}_{n\hspace{0.17em}-\hspace{0.17em}1}{x}^{n\hspace{0.17em}-\hspace{0.17em}1}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}\cdots \hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}{a}_0$$ (15.1)

where $a_n\hspace{0.17em}\ne \hspace{0.17em}0$ and n is a positive integer or zero is called a polynomial function. ...