Table of Contents
In this chapter, we focus on what an algebraic expression is and how it is built and interpreted. The broader ideas of “variables and expressions” are introduced in the parent chapter; here we look specifically at the structure and language of algebraic expressions.
What an algebraic expression is
An algebraic expression is a combination of:
- numbers,
- variables (like $x$, $y$, $a$),
- operation symbols ($+$, $-$, $\times$, $\div$),
- and sometimes parentheses,
written according to the rules of algebra, but without an equals sign.
Examples of algebraic expressions:
- $x + 3$
- $5y - 2$
- $2a + 3b$
- $7(x - 4)$
- $\dfrac{m}{2}$
- $4p^2$ (this involves an exponent and will be connected to powers in a later chapter)
Expressions are like “mathematical phrases.” They do not make a statement like “equals 10”; they just describe a quantity.
By contrast, $x + 3 = 10$ is not an expression; it is an equation (which is treated in a later chapter).
Parts of an algebraic expression
Many algebraic expressions can be broken into smaller pieces. The basic parts you should recognize are:
- Term: A piece of an expression that is added or subtracted.
- Coefficient: The numerical factor in front of a variable.
- Variable part: The letter(s) representing an unknown or changing quantity.
- Constant term: A term with no variable, just a number.
Consider the expression:
$$
4x + 7
$$
- Terms: $4x$ and $7$
- Coefficient of $x$: $4$
- Variable part: $x$
- Constant term: $7$
Another example:
$$
3a - 5b + 2
$$
- Terms: $3a$, $-5b$, and $2$
- Coefficient of $a$: $3$
- Coefficient of $b$: $-5$
- Constant term: $2$
Note that we treat subtraction as adding a negative term. So in $3a - 5b + 2$, the term with $b$ is $-5b$.
Like terms (concept only, no simplification yet)
Within expressions, some terms have the same variable part. These are called like terms.
- In $4x + 7x$, the terms $4x$ and $7x$ are like terms: both are “something times $x$.”
- In $3a - 5a + 2$, the terms $3a$ and $-5a$ are like terms.
- In $2x + 3y$, the terms $2x$ and $3y$ are not like terms because one has $x$ and the other has $y$.
The idea of like terms is important when simplifying expressions (which will be developed more fully when working with operations on expressions).
Typical forms of algebraic expressions
Algebraic expressions can take several common shapes. Recognizing these forms helps you read and write them correctly.
1. Sum or difference expressions
These are expressions where quantities are added or subtracted.
Examples:
- $x + 5$
- $10 - y$
- $3m + 2n - 4$
Each separate piece joined by $+$ or $-$ is a term.
2. Product expressions
These involve multiplication between numbers and variables, or between variables.
Examples:
- $4x$ (means $4 \times x$)
- $ab$ (means $a \times b$)
- $7(x - 2)$ (means $7$ times the quantity $x - 2$)
In algebra, we usually omit the multiplication sign between a number and a variable or between variables. Instead of writing $4 \times x$, we write $4x$.
3. Quotient expressions
These involve division written as a fraction.
Examples:
- $\dfrac{x}{3}$
- $\dfrac{a + 1}{b}$
- $\dfrac{m}{2n}$
A fraction bar represents division. For example, $\dfrac{x}{3}$ means $x \div 3$.
4. Expressions with parentheses
Parentheses group parts of an expression to show which operations go together.
Examples:
- $3(x + 2)$
- $(a - 1)(b + 4)$
- $\dfrac{2(x - 5)}{3}$
Parentheses change how you interpret the expression and which operations are done first.
5. Expressions with exponents (briefly)
Sometimes expressions include exponents, like:
- $x^2$
- $3y^3$
- $2a^2 + 5$
Here $x^2$ means $x \times x$, and $y^3$ means $y \times y \times y$. The detailed study of powers and exponents appears later; here, just recognize that they are also part of algebraic expressions.
Translating verbal phrases into algebraic expressions
Many problems describe quantities in words. You often need to translate a phrase into an algebraic expression. The exact solving of word problems will be treated later; here we focus on the translation into expressions.
Common word patterns and their algebraic forms (let $x$ represent some number):
- “A number plus 5” $\rightarrow x + 5$
- “5 more than a number” $\rightarrow x + 5$
- “A number minus 3” $\rightarrow x - 3$
- “3 less than a number” $\rightarrow x - 3$
- “The sum of a number and 7” $\rightarrow x + 7$
- “The difference between a number and 2” $\rightarrow x - 2$
- “Twice a number” $\rightarrow 2x$
- “Three times a number” $\rightarrow 3x$
- “Half of a number” $\rightarrow \dfrac{x}{2}$
- “The product of 4 and a number” $\rightarrow 4x$
- “The quotient of a number and 5” $\rightarrow \dfrac{x}{5}$
Some phrases are a little more complex:
- “5 more than twice a number”
Twice a number: x$
5 more than that: x + 5$ - “3 less than the product of 4 and a number”
Product of 4 and a number: x$
3 less than that: x - 3$ - “The sum of a number and 2, all divided by 3”
Sum of a number and 2: $x + 2$
All divided by 3: $\dfrac{x + 2}{3}$
Notice how parentheses or fraction bars are used to show which part of the phrase “goes together.”
Writing expressions for simple situations
You often use algebraic expressions to describe real situations. The details of solving will appear in later chapters; here we only set up expressions.
Examples:
- “The total cost of $x$ notebooks if each notebook costs \$2.”
- Cost of one notebook: $2$
- Cost of $x$ notebooks: $2x$
- “Your age in $y$ years if you are now 12 years old.”
- Current age: $12$
- In $y$ years: $12 + y$
- “The length of a ribbon after cutting off $x$ centimeters from a 50-centimeter ribbon.”
- Original length: $50$
- After cutting off $x$: $50 - x$
- “The perimeter of a rectangle with length $l$ and width $w$.”
- Perimeter: $2l + 2w$
Each expression captures a relationship between quantities using variables and numbers.
Common notational conventions in expressions
When reading and writing algebraic expressions, some stylistic rules are standard:
- Multiplication:
- Between a number and a variable: write $4x$, not $4 \times x$.
- Between variables: write $ab$, not $a \times b$.
- Between parentheses: write $(x + 1)(x - 2)$.
- Division:
- Often shown with a fraction: $\dfrac{x}{3}$ instead of $x \div 3$.
- When the entire numerator or denominator is a sum or difference, use parentheses or a fraction bar:
$\dfrac{x + 4}{2}$ means $(x + 4) \div 2$. - Order of factors:
- Usually write the numerical coefficient first, then the variable: $3x$, not $x3$.
- If there are several variables, a common order is alphabetical: $2ab$, not $2ba$ (though both mean the same mathematically).
These conventions make expressions easier to read and less confusing.
Interpreting the meaning of an expression
Finally, it is useful to read an expression and say in words what it represents.
Examples:
- $5x$ could be read as “five times a number $x$” or “the product of 5 and $x$.”
- $x + 3$ could be read as “a number $x$ increased by 3” or “the sum of $x$ and 3.”
- $7(x - 2)$ could be read as “seven times the quantity $x$ minus 2.”
- $\dfrac{h}{2}$ could be read as “half of $h$.”
- $3n + 4$ could be read as “three times a number $n$ plus 4.”
Being able to move comfortably between words and algebraic expressions is a central skill for all later algebra topics.