Polynomial Expressions

Introduction

In mathematics, a polynomial is an expression consisting of variables and coefficients, where the exponents are non-negative integers.

Polynomials can also be classified by their degree, which is the sum of the variable exponents in any one term.

Polynomial expressions have the following operations: addition, subtraction, multiplication, division.

Polynomials have many applications; here are a few examples of some applications:

Planetary Motion : Johannes Kepler (1571-1630) discovered that the orbits of planets in our solar system follow paths around the sun that are ellipses. Kepler discovered several laws about planetary motion, one of which is described below.
Pierre Fermat (1601-1665) conjectured that every number of the form $F_{n} = 2^{2^{n}} + 1$ $LaTeX: F_n=2^{2^n}+1$ where n is a nonnegative integer, is prime. These numbers are called Fermat numbers in his honor.
The following formula gives the monthly cost M of a mortgage (a loan taken out with real estate as collateral) if the principal (the amount borrowed) is p dollars, which is to be totally repaid in n months, and the monthly interest rate (as a decimal) is r: $LaTeX: M=\frac{rp}{1-\left(1+r\right)^{-n}}$ $M = \frac{r p}{1 - {(1 + r)}^{- n}}$
Constructing a Pyramid
For a class project, you and your classmates are constructing two pyramids using aluminum cans. One of your pyramids will have a triangular base and the other will have a square base as shown below.

images of two pyramids of cans; one with a triangular base and the other with a square base

The formula for the total number of cans C in a triangular-base pyramid with n layers is $Formula for total number of cans$ . For a square-base pyramid, it is equation image indicator $C = \frac{1}{3} n^{3} + \frac{1}{2} n^{2} + \frac{1}{6}$ .

Economic Models

Economic models are used by both large and small businesses. Suppose you work in the marketing department of a worldwide manufacturer and your company has recently released a new product on the market. The demand function p shown at the right for the new product is equation image indicator LaTeX: p=34-2x^2 $p = 34 - 2 x^{2}$ where x is the number of units produced in millions.

Essential Questions

What are Polynomials?
How are Polynomials classified by type and degree?
How are Polynomials added, subtracted, multiplied, and divided?
What is Pascal's Triangle and how it is used in multiplying Polynomials?
How are Polynomials expressions divided using Synthetic Division?
How are Polynomials expressions divided using Long Division?
How can polynomials be used to express realistic situations?

Key Terms

The following key terms will help you understand the content in this module.

Algebra - The branch of mathematics that deals with relationships between numbers, utilizing letters and other symbols to represent specific sets of numbers, or to describe a pattern of relationships between numbers.

Binomial Expression - An algebraic expression with two unlike terms.

Coefficient - A number multiplied by a variable.

Constant Term - A quantity that does not change its value.

Expression - A mathematical phrase involving at least one variable and sometimes numbers and operation symbols.

Factor - When two or more integers are multiplied, each integer is a factor of the product. "To factor" means to write the number or term as a product of its factors.

Integer - The set of numbers ..., –3, –2, –1,0,1,2,3,...

Irrational Number - A number whose decimal form is nonterminating and nonrepeating. Irrational numbers cannot be written in the form a/b, where a and b are integers (b cannot be zero). So all numbers that are not rational are irrational.

Long Division - Is an algorithm for dividing a polynomial by another polynomial of the same or lower degree.

Monomial Expression - An algebraic expression with one term.

Pascal's Triangle - A number triangle with numbers arranged in staggered rows such that $LaTeX: a_{nr}=\frac{n!}{r!\langle n-r\rangle!}=\frac{n}{r},$ where equation image indicator $(\frac{n}{r})$ is a binomial coefficient.

Polynomial - The sum or difference of two or more monomials.

Polynomial Function - A polynomial function is defined as a function, $LaTeX: f\left(x\right)=a_0x^n+a_1x^{n-1}+a_2x^{n-2}+...+a_{n-2}x^2+a_{n-1}x+a_n$ where the coefficients are real numbers.

Rational Number - A number expressible in the form a/b or –a/b for some fraction a/b. The rational numbers include the integers.

Standard Form of a Polynomial - To express a polynomial by putting the terms in descending exponent order.

Synthetic Division - Is a shorthand, or shortcut, method of polynomial division in the special case of dividing by a linear factor - and it only works in this case.

Term - A number, a variable, or a product of numbers and variables.

Trinomial - An algebraic expression with three unlike terms.

Variable - A letter or symbol used to represent a number.

Whole Numbers - The numbers 0, 1, 2, 3, ....

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