Division: Polynomials and Binomials
Division of polynomials and binomials is a fundamental concept in algebra that focuses on how these mathematical expressions can be divided by each other. A polynomial is defined as a mathematical expression consisting of variables and coefficients, while a binomial specifically refers to a polynomial with two terms. The division can be executed using two primary methods: polynomial long division and synthetic division.
Polynomial long division resembles the long division process used in basic arithmetic, where the dividend and divisor are placed in a specific format, and the leading terms are used to derive the quotient and remainder sequentially. On the other hand, synthetic division offers a more efficient shortcut when the divisor is a binomial of the form (x - c), where c is a constant. This method simplifies the division process by using a systematic pattern of multiplication and addition.
Understanding these division techniques is crucial for various applications in mathematics, including finding the zeros of polynomial functions, which are essential for graphing and analyzing polynomial behavior. Ultimately, mastering polynomial division equips learners with valuable skills for tackling more complex algebraic problems and concepts.
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Subject Terms
Division: Polynomials and Binomials
A polynomial is a mathematical expression consisting of variables and their coefficients and involving only addition, subtraction, multiplication, and non-negative exponents; a binomial is a polynomial which is the sum of two monomial terms. A polynomial can be divided by a binomial using long division.
An nth degree polynomial (or polynomial of degree n) in one variable is an algebraic expression of the form
where is a natural number, and are complex numbers, and . A special case of is a binomial, which is a polynomial with two terms. For example, is a 3rd degree polynomial, and and are binomials of degree and , respectively.
If a function is given by , with the same conditions as , then is an nth degree polynomial function. The division algorithm states that if an nth degree polynomial function is divided by a nonzero polynomial function of degree , then there exist unique polynomial functions and such that
where either the degree of is less than that of , or . In , is the dividend, is the divisor, is the quotient, and is the remainder. Examples are given below.
The objective of this article to introduce two methods for finding the functions and in (2): polynomial long division and synthetic division. The scope of this article is limited to division of a polynomial by a binomial.
Overview
Polynomial long division is set up and carried out in a manner similar to that of its arithmetic counterpart. For example when dividing by , the divisor and dividend are placed on the outside and inside, respectively, of a long division symbol, as shown below.
The terms of the quotient are generated by dividing the leading term of the dividend by the leading term of the divisor, multiplying the result by the divisor, and subtracting to obtain a polynomial whose degree is less than that of the dividend. Subsequent terms of the quotient are generated by repeating the process with the resulting polynomials until a polynomial whose degree is less than that of the divisor is obtained. Our example thus begins with dividing by , and continues as described above. This is shown below with the quotient displayed on top of the division sign.
The polynomial in the last line is the remainder. This can be rewritten as follows.
Synthetic division of polynomials is a shortcut that can be taken whenever the divisor is of the form , where is a constant. If the dividend is of the form , then the synthetic division of the dividend by is set up as follows.
The pattern is to multiply (by ) diagonally, and add vertically, as shown below.
The first numbers in the list at the bottom are the coefficients of the quotient in descending order, and the number is the remainder . As an example, the synthetic division of by is shown below.
This can be rewritten as .
A major topic in calculus is curve sketching, in particular sketching graphs of polynomial functions. In order to do this, it is essential to be able to find zeros of polynomial functions. For example, the x-intercepts of the graph of the function correspond to the zeros of the function. is a zero of , so is a factor. Synthetic division can be used to find the other factors as follows.
So , the zeros of are and , and the x-intercepts of the graph of are and . This can be seen in Figure 1.
Bibliography
Angel, Allen R., and Dennis C. Runde. Elementary Algebra for College Students. Boston: Pearson, 2011. Print.
Lial, Margaret L., E J. Hornsby, and Terry McGinnis. Intermediate Algebra. Boston: Pearson, 2012. Print.
Miller, Julie. College Algebra Essentials. New York: McGraw, 2013. Print.
Sullivan, Michael. Precalculus : Enhanced with Graphing Utilities. Upper Saddle River: Pearson, 2013. Print.
Young, Cynthia Y. College Algebra. Hoboken: Wiley, 2012. Print.