Algebraic Expressions

Introduction

Algebraic expressions are made up of terms. A term is a constant or the product of a constant and one or more variables. Some examples of terms are $7,y,5x^{2},9a,$ and $13xy$ . The constant that multiplies the variable(s) in a term is called the coefficient. We can think of the coefficient as the number in front of the variable. For example, the coefficient of the term 3x is 3. When we write x, the coefficient is 1, since x = 1(x). To evaluate an algebraic expression means to find the value of the expression when the variable is replaced by a given number. To evaluate an expression, we substitute the given number for the variable in the expression, and then simplify the expression using the order of operations.

Some terms share common traits. Terms are called like terms if they have the same variables and exponents. All constant terms are also like terms. Consider the terms $5x,7,n^{2},4,3x,9n^{2},-2xy$ , and $8xy$ . From this list, 7 and 4 are like terms, 5x and 3x are like terms, $-2xy$ and $8xy$ are like terms, and $n^{2}$ and $9n^{2}$ are like terms.

We can simplify an expression by combining the like terms. To do this, we add the coefficients and keep the same variable. For example, say we want to simplify the expression $x^{2}+4x^{2}-3x+6+x$ . The coefficients of the like terms $x^{2}$ and $4x^{2}$ are 1 and 4, and add up to 5. The coefficients of the like terms -3x and x are 1 and -3, and add to -2. 6 is not like to any of the other terms, so the constant term remains as 6. So, we can combine like terms and get that $x^{2}+4x^{2}-3x+6+x=(x^{2}+4x^{2})+(-3x+x)+6=5x^{2}-2x+6$ by adding up the coefficients of like terms.