Polynomials are one of the most frequently studied objects in mathematics. It is no surprise, then, that we devote lengthy chapters to them in both Algebra I and Algebra II. This chapter focuses primarily on the roots or zeros of polynomials, and, in the process, on division of polynomials by binomials.
The first section introduces a new form of a polynomial: nested form. Nested form is useful when evaluating polynomial functions by hand. This section explains how to convert a polynomial function to nested form and how to use nested form to evaluate a polynomial function for any value of the variable.
The next section explains how to divide a polynomial by a binomial using long division. This is the same long division learned in grade school, but with a variable in the divisor instead of a constant. This section also introduces a shortcut for finding the remainder when a polynomial is divided by a binomial: the Remainder Theorem. The Factor Theorem, which follows from the Remainder Theorem, provides an easy way for determining whether a given binomial is a factor of a given polynomial.
Since long division can be time-consuming, mathematicians have developed an easier way to divide a polynomial by a binomial. This method is called synthetic division. Synthetic division is similar to computing the value of a polynomial function in nested form, and it provides additional information. In addition to giving the remainder when a polynomial function is divided by a binomial x - a--the value of P(a)--synthetic division also yields the quotient when P(x) is divided by x - a. This process is discussed in detail in section three.
The subsequent section is about a specific use of synthetic division--finding the roots of a polynomial function. This section explains how to find all the rational roots of a polynomial function, using the Rational Zeros Theorem. The final section in this chapter deals with the complex roots of an equation, and introduces two new theorems. These are the Conjugate Zeros Theorem and the Fundamental Theorem of Algebra.
As the name of the theorem implies, polynomial functions and their roots are fundamental to the study of algebra. An entire branch of algebra is devoted solely to examining polynomials and their roots, and the material covered in this chapter is a jumping-off point for more elaborate study. Polynomials should be studied both because they are one of the most frequently discussed objects in mathematics and because they are one of the most interesting.
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