A major part of any math course is learning algebra. You may not have realized it already, but you’ve already been using algebraic techniques to answer many of the trigonometry questions posed in previous lessons. For example, any time you want to solve for an unknown, some algebraic technique is being used. It turns out that no field of mathematics is mutually exclusive with algebra, so it makes sense we continue this course with an introduction to algebra next. This lesson will introduce you to a variety of terms and phrases that will be used in this unit and onward.
An algebraic expression is one containing only algebraic symbols and operations (addition, subtraction, multiplication, division, roots, and powers). For example:
… is an algebraic expression containing three terms. The plus and the minus signs divide an expression into terms.
- If the expression contains trigonometric, exponential, or logarithmic functions, it’s called a transcendental expression. For the sake of simplicity, we’ll refer to all expressions as algebraic.
Another example of an expression is shown below:
The x is considered a variable, which is an unknown placeholder that can be modified to whatever number of expression you set it to. For example, setting x = 1, we get: 2 ( 1 ) ² + 5 ( 1 ) + 3 ⇒ 2 + 5 + 3 ⇒ 10. However, if this expression were an equation (difference explained below), the variable would have a specific meaning, given that equations are solvable.
As mentioned, equations differ from expressions. When we use the word equation, we are suggesting that the algebraic expression is solvable for its unknown variables. An indicator of whether a mathematical statement is an equation or an expression is the presence of the equal sign. Examples of equations are:
- 2x² + 5x + 3 = 0 (same as the expression above except now it’s equal to 0)
- 6x – 4 = x + 1
- y = 3x – 5
Constants and Coefficients
Let’s look back at the original expression:
We’ve established three things so far about this statement:
- It’s an algebraic expression as opposed to an equation
- It contains three terms
- It possesses the variable x
The last thing you need to know is that the term, + 3, is considered a pure constant. Technically, a constant describes any number in an algebraic expression (including the –2 in front of the x and the +1 in front of x²), but for simplicity sake, we’ll refer to “constants” as terms that are exclusively numbers, such as + 3.
A constant is a quantity that does not change in value in a particular problem.
A variable is a quantity that may change during a particular problem.
So, what are the constant(s) in this expression? null
5, though technically 3 and 4 could also be included but they get a special name instead (detailed below).
When a term is made up of a constant multiplied by a variable or variables, that constant is called a coefficient. Using the expression 3x² + 4x + 5, the term 3x² has a coefficient of 3 while 4x has a coefficient of 4. Therefore, the coefficient of a term is the constant part of the term, and is usually written before the variable part of the term. Another important thing to know is if a coefficient in front of a term is missing, it is automatically 1.
Degree of a Term
The degree of a term refers to the integer power to which the variable is raised. For example, in the example above, the term 3x² is a second-degree term. 4x is a first-degree term because the x, while not shown, has an exponent of 1. Other examples:
- 5y9 has a degree of _______
- 4x³y9 has a degree of _______ (tricky)
More on how to determine the degree of a term is explained in the video below:
Types of Expressions
A monomial is an algebraic expression with a single term. Examples include:
A binomial is an algebraic expression with two terms. Examples include:
Two or more terms are collectively referred to as polynomials (or multinomial). Keep in mind that the variables in a polynomial expression must have positive exponents. The first example below is a polynomial while the second and third are not.
The second expression contains y raised to a power of negative 2, while the last example has x² in the denominator, which is the same as x raised to a power of negative 2 in the numerator. This will be further explained when discussing negative exponents.