Variables form the foundation of Algebra. They represent unknown values in expressions and equations. Letters are used to symbolize these variables and differentiate them from numbers. Sometimes a problem will give you variables with which to work. Other times, as in the case of a word problem, you must come up with your own letters to represent unknown information.
The two building blocks of algebra, numbers and variables, come together in one of two ways--either as an expression or an equation. An expression does not contain an equals sign and cannot be used to find the missing value of a variable. "2X - 4" is an expression. On the other hand, an equation does include an equals sign and, therefore, describes a relationship between the numbers and variables that can possibly be used to solve for the variables. "2X - 4 = 8" is an equation.
A monomial is a special type of expression or equation. To qualify as a monomial, an expression or equation has to meet three rules: It cannot contain addition or subtraction because "mono" indicates that it can include only one term; it cannot have any variables under a division bar (numbers are okay); and it cannot have any negative exponents or fractional exponents. For example, 3X^2 would be a monomial. 4a^-3 or 2/Y would not be monomials.
The word "polynomial" contains the root "poly" which means "many" because a polynomial is a chain of more than one monomial added to, or subtracted from, one another. Because it is a type of expression/equation, a polynomial contains variables. Each term being added or subtracted in the equation or expression has to pass the monomial test for it to qualify as a polynomial. "2X^2 - 4x + 8" would be a polynomial because 2x^2, 4x, and 8 all pass as monomials.
To work with the variables in a polynomial, you must know a few basic skills and terms. To arrange a polynomial in ascending or descending order, you must calculate the total value of all of the powers in each monomial "piece." Then, place them in order from highest to lowest or lowest to highest. For instance, 2X^3 + 5X^2Y^3 - 4y^2 placed in descending order would be 5X^2Y^3 + 2X^3 - 4y^2. The "degree" of the polynomial refers to the highest total exponent value in a single monomial, or term. In this case, the polynomial would be of the 5th degree. Finally, the "leading coefficient" describes the number in front of the first term when they are placed in descending order. Hence, the leading coefficient of this expression would be "5."