Algebra: Variables and Polynomials
Variables and Polynomials
In this Algebra video tutorial, we explore the various usages of variables and define the domain of a variable, monomials, and polynomials. To prepare for the coming lessons, we briefly touch on some of the naming conventions for different types of variables in mathematics.
Variables are quantities that can change or vary. We use letters as placeholders to represent a varying quantity. Each variable in mathematics is associated with a set of values that the variable can take on; this set of values is called the domain of the variable.
Frequently, variables are used in mathematics without explicitly stating what the domain of each variable is. In such cases, the domain is assumed to be obvious and should be clearly understood. When a domain is assumed, it typically follows the common conventions that are used in mathematics for naming variables. For example, when we see the variable x or y used, we can be certain that the domain for both is the real numbers, unless otherwise specified. Some common conventions are shown below.
When we work with variables, we typically want to solve an equation, The subset of the domain that solves an equation is called the solution set. The solution set will vary depending on the domain. So, the domain is very important. For example, for the variable x with its domain as the real numbers, the equation below has the solution 4/3. However, if the domain is the integers, then the solution set is the empty set, since the equation has no solution in the domain.
Variables are used in many settings. So, it is important to understand how a variable is used in a particular context. For example, there are linear equations that involve x and y as real number variables like these: 3x + 5y + 4 = 0 and 2x + 3y - 1 = 0. These are sets of points in the plane. However, we can add a level of abstraction and talk about the set of all such sets of the form Ax + By + C = 0, where A, B, and C are real variables.
We can combine variables into larger expressions. A monomial is any product of real numbers and variables with integer exponents like 4xy, -.5, y. The degree of a monomial is the sum of the exponents of all of the variables in the product. A polynomial is any sum of monomials. The degree of a polynomial is the largest degree of its monomial terms.