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# Algebraic Expressions

by Ron Kurtus (revised 18 January 2022)

An * algebraic expression *is a combination of numbers, variables, constants, operators, and parentheses or grouping brackets that are stated as a single entity.

Some expressions are made up of sub-expressions.

Much of Algebra concerns simplifying expressions to facilitate the solution of equations.

Questions you may have include:

- What are some examples of expressions?
- What are sub-expressions?
- How do you simplify expressions?

This lesson will answer those questions.

## Examples of expressions

An expression consists of a combination of numbers, variables, constants, operators and/or parentheses. Examples of expressions include:

3

x

2y − 3

x^{2}+ (4x + 2)/2

25 + (x + 3)^{3}/2z − [5 + z(x+ 1) − 7]

### An operator alone is not an expression

Although an individual number, variable, or constant can be an expression, an operator by itself is not an expression. In other words, **3**, **x** and **c** can be expressions, but **+** and **÷** are **not** expressions.

### Expressions are used in equations

An equation consists of an expression on each side of the equals sign.

The equation** x ^{2} + (3y + 2)/2 = 7z** consists of the expressions

**x**and

^{2}+ (3y + 2)/2**7z**.

## Monomial and other expressions

An expression can be defined by the number of plus (+) or minus (−) signs it has, after it has been simplified.

### Monomial

A monomial expression has one term and **no** plus or minus signs. Examples of monomials include:

2

7x

xy^{2}z

### Binomial

A binomial expression has two terms, with at least one variable or constant, and one plus (+) or minus (−) sign. Examples of binomials include:

x + 2

7x − 3y

xy^{2}z/2 + 8

But note that **5x − 3x** is not a binomial, because it simplifies to **2x**.

A trinomial expression has three terms, with at least one variable or constant, and two plus (+) or minus (−) signs. Examples of trinomials include:

ax + y − 7

x + 2y + z

xy^{2}z/2 − z + 8

Note that **x ^{3} − 15 + 8**

**x**is not a trinomial. It is really a binomial, since it can be simplified to:

^{3}**9x**

^{3}− 15## Sub-expressions

An expression can consist of several expressions or sub-expressions.

For example, the expression **2y − 3** consists of sub-expressions **2y** and **3**. Also **2y** consists of sub-expressions **2** and **y**.

Likewise, **x ^{2} + 2x + 1** consists of sub-expressions

**x**,

^{2}**2x**, and

**1**. And

**x**consists of sub-expression

^{2}**x**(since

**x**is

^{2}**x**times

**x**). And

**2x**consists of sub-expressions

**2**and

**x**.

### Exercise example

What are the expressions in the equation** y + 1 = x ^{2} + x(y + 1)**?

Expressions on each side of the equation are

y + 1andx^{2}+ x(y + 1)Expressions in

y + 1areyand1Expressions in

xare^{2}+ x(y + 1)xand^{2}x(y + 1)Expression in

xis^{2}xExpressions in

x(y + 1)arexand(y + 1)Expressions in

(y + 1)areyand1

## Simplifying expressions

Expressions are meant to be used in equations. Thus, the expressions should be meaningful and in their simplest form to facilitate solving the equation. Much of Algebra concerns putting expressions in meaningful forms.

For example, the expression, **x ^{2} + (4x + 2)/2** can be reduced to

**x**by completing the division by

^{2}+ 2x + 1**2**in the expression

**(4x + 2)/2**.

In some cases, it is useful to put an expression in a less simplified form, to facilitate the solution of an equation. For example, **x ^{2} + 2x** could be factored into

**x(x + 2)**to make it easier to solve the equation

**x**.

^{2}+ 2x = 0## Summary

An expression is a combination of numbers, variables, constants, operators and parentheses or grouping brackets that are stated as an entity. An operator or grouping bracket alone is not an expression. An equation consists of an expression on each side of the equals sign. Some expressions are made up of sub-expressions.

Much of Algebra concerns simplifying expressions to facilitate the solution of equations.

Be expressive when explaining things

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## Algebraic Expressions