# 4.1 Algebraic expressions

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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. Operations with algebraic expressions and numerical evaluations are introduced in this chapter. Coefficients are described rather than merely defined. Special binomial products have both literal and symbolic explanations and since they occur so frequently in mathematics, we have been careful to help the student remember them. In each example problem, the student is "talked" through the symbolic form.Objectives of this module: be familiar with algebraic expressions, understand the difference between a term and a factor, be familiar with the concept of common factors, know the function of a coefficient.

## Overview

• Algebraic Expressions
• Terms and Factors
• Common Factors
• Coefficients

## Algebraic expression

An algebraic expression is a number, a letter, or a collection of numbers and letters along with meaningful signs of operation.

## Expressions

Algebraic expressions are often referred to simply as expressions , as in the following examples:

$x+4$ is an expression.

$7y$ is an expression.

$\frac{x-3{x}^{2}y}{7+9x}$ is an expression.

The number 8 is an expression. 8 can be written with explicit signs of operation by writing it as $8+0$ or $8\cdot 1$ .

$3{x}^{2}+6=4x-1$ is not an expression, it is an equation . We will study equations in the next section.

## Terms

In an algebraic expression, the quantities joined by $"+"$ signs are called terms.

In some expressions it will appear that terms are joined by $"-"$ signs. We must keep in mind that subtraction is addition of the negative, that is, $a-b=a+\left(-b\right)$ .

An important concept that all students of algebra must be aware of is the difference between terms and factors .

## Factors

Any numbers or symbols that are multiplied together are factors of their product.

Terms are parts of sums and are therefore joined by addition (or subtraction) signs.
Factors are parts of products and are therefore joined by multiplication signs.

## Sample set a

Identify the terms in the following expressions.

$3{x}^{4}+6{x}^{2}+5x+8$ .

This expression has four terms: $3{x}^{4},6{x}^{2},\text{\hspace{0.17em}}5x,$ and 8.

$15{y}^{8}$ .

In this expression there is only one term. The term is $15{y}^{8}$ .

$14{x}^{5}y+{\left(a+3\right)}^{2}$ .

In this expression there are two terms: the terms are $14{x}^{5}y$ and ${\left(a+3\right)}^{2}$ . Notice that the term ${\left(a+3\right)}^{2}$ is itself composed of two like factors, each of which is composed of the two terms, $a$ and 3.

${m}^{3}-3$ .

Using our definition of subtraction, this expression can be written in the form ${m}^{3}+\left(-3\right)$ . Now we can see that the terms are ${m}^{3}$ and $-3$ .

Rather than rewriting the expression when a subtraction occurs, we can identify terms more quickly by associating the $+$ or $-$ sign with the individual quantity.

${p}^{4}-7{p}^{3}-2p-11$ .

Associating the sign with the individual quantities we see that the terms of this expression are ${p}^{4},\text{\hspace{0.17em}}-7{p}^{3},\text{\hspace{0.17em}}-2p,$ and $-11$ .

## Practice set a

Let’s say it again. The difference between terms and factors is that terms are joined by signs and factors are joined by signs.

List the terms in the following expressions.

$4{x}^{2}-8x+7$

$4{x}^{2},\text{\hspace{0.17em}}-8x,\text{\hspace{0.17em}}7$

$2xy+6{x}^{2}+{\left(x-y\right)}^{4}$

$2xy,\text{\hspace{0.17em}}6{x}^{2},\text{\hspace{0.17em}}{\left(x-y\right)}^{4}$

$5{x}^{2}+3x-3x{y}^{7}+\left(x-y\right)\left({x}^{3}-6\right)$

$5{x}^{2},3x,-3x{y}^{7},\text{\hspace{0.17em}}\left(x-y\right)\left({x}^{3}-6\right)$

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