The coefficient of a term is the constant factor that multiplies the variable portion of the term. It can be thought of as the numerical value situated in front of the variable. For instance, in the term $$3x$$, the coefficient is $$3$$. When a variable is written by itself without an explicit number, such as $$x$$, its coefficient is implicitly $$1$$, because $$x$$ is equivalent to $$1 \cdot x$$.

Coefficient of a Term

Terms in algebraic expressions are referred to as **like terms** when they are either constants or have the exact same variables raised to the identical powers. These terms share common traits that allow them to be grouped. For instance, $$5x$$ and $$3x$$ are like terms. Similarly, $$n^2$$ and $$9n^2$$ are like terms. The constants $$7$$ and $$4$$ are also like terms.

Like Terms in Algebra

To identify the terms within an algebraic expression, locate the parts of the expression separated by addition ($+$) or subtraction ($-$) signs. Each separated part is a single term.

- In $9x^2 + 7x + 12$, the addition signs divide the expression into three terms: $9x^2$, $7x$, and ${12}$.
- In $8x + 3y$, the addition sign separates two terms: $8x$ and $3y$.

A standalone constant such as ${12}$ qualifies as its own term, and terms with different variable parts—like $9x^2$ (involving $x$ squared) versus $7x$ (involving $x$ to the first power)—are counted as distinct terms.

Identifying Terms in an Expression

A monomial is an algebraic expression that consists of exactly one term. A specific case is a monomial in one variable, which takes the form $$ax^m$$, where $$a$$ is a constant and $$m$$ is a whole number. Examples of monomials in one variable include $$2x$$, $$17z$$, and $$4y^2$$. However, monomials can also contain multiple variables, such as $$5abc$$ or $$-4a^2b^3c^2$$. Because they represent a single mathematical unit, monomials serve as the fundamental building blocks of polynomials.

Monomial

A project manager uses the expression 45h + 200 to calculate the total cost of a service, where 45h is the labor cost and 200 is the equipment fee. What is the mathematical name for the individual parts, such as 45h and 200, that are added together to form this expression?

Identifying Components of a Service Cost Formula

Based on the provided scenario, define the mathematical concept of an 'algebraic term' and identify the individual terms present in the startup's formula. Briefly state why each component you identified qualifies as a term according to your definition.

Analyzing Production Cost Components

The degree of a term is the sum of the exponents of all its variables. For a term with a single variable, the degree equals the exponent on that variable — for example, $$8y^2$$ has a degree of $$2$$ and $$-13a$$ has a degree of $$1$$ (since $$a = a^1$$). For a term with multiple variables, the exponents are added together — for example, $$-9x^3y^5$$ has a degree of $$3 + 5 = 8$$. A special case is a constant term (a monomial with no variable at all), which always has a degree of $$0$$ because there are no variable factors present.

Degree of a Term

In the expression $$25w + 15$$, what are $$25w$$ and $$15$$ called?

What can an algebraic term be?

List the terms of $$9a + y + 7$$ in order from left to right.

Which of the following is NOT a single algebraic term?

Question:
What is an algebraic term? Give three examples.

Sample answer:
An algebraic term is a constant or the product of a constant and one or more variables. Examples are $$7$$, $$y$$, and $$5x^2$$.

Key points:
- A term can be a constant.
- A term can be a constant multiplied by one or more variables.
- Three valid algebraic terms are included.

Rubric:
The answer correctly defines an algebraic term and gives three valid examples.

What is an algebraic term? Give three examples.

Match each algebraic term to its description.

The constant $$500$$ is an algebraic term.

In the expression $$7 + 5x$$, each part joined by the plus sign is a(n) _____.

An algebraic **term** is a constant, or the product of a constant and one or more variables. They are the fundamental building blocks of algebraic expressions. Examples of terms include $$7$$, $$y$$, $$5x^2$$, $$9a$$, and $$b^5$$.

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An **expression** in algebra is a number, a variable, or any combination of numbers and variables joined by operation symbols such as $$+$$, $$-$$, $$\cdot$$, and $$\div$$. Expressions represent a value but do not make a complete mathematical statement on their own. For example, $$3 + 5$$, $$n - 1$$, $$6 \cdot 7$$, and $$\frac{x}{y}$$ are all expressions. A single number like $$7$$ or a single variable like $$x$$ also qualifies as an expression.

Algebraic Expression

https://openstax.org/details/books/elementary-algebra-2e

OpenStax Elementary Algebra

https://openstax.org/details/books/intermediate-algebra-2e

OpenStax Intermediate Algebra

An **algebraic equation** is a complete mathematical statement formed by linking two algebraic expressions with an equal sign ($$=$). The equal sign acts as the verb of the sentence, asserting that the expressions on either side hold the same value. For example, $$3 + 5 = 8$$ and $$y + 9 = 2y - 3$$ are equations because they make a complete, verifiable claim about equality. This structure directly parallels a complete sentence in English, where a subject and verb express a full thought.

Algebraic Equation

Algebraic Term

Exponential notation is a concise mathematical method used to represent a number or variable that is repeatedly multiplied by itself. It consists of two parts: a **base** and an **exponent**. The base is the quantity being multiplied, and the exponent, written as a superscript, indicates the exact number of times the base is used as a factor. For example, in the expression $$a^m$$, the exponent $$m$$ tells us how many times we use the base $$a$$ as a factor, condensing the expanded form into a shorter notation.

Exponential Notation in Algebra

To simplify an expression is to perform all possible mathematical operations within it until no further arithmetic can be done. For instance, simplifying the expression $$4 \cdot 2 + 1$$ requires evaluating the multiplication first to get $$8$$, and then performing the addition to arrive at a final value of $$9$$. A best practice when simplifying is to organize the work vertically, writing each successive step clearly on a new line directly below the previous one. Furthermore, equal signs should be omitted between these steps during simplification; avoiding equal signs prevents misleadingly turning the expression into an equation and thereby confusing the fundamentally distinct concepts of simplifying expressions and solving algebraic equations.

Simplifying an Algebraic Expression

As part of an initiative to standardize technical language in your department, you have been asked to write a brief entry for the company's 'Math Standards' wiki. Your task is to define the term 'algebraic expression.' In your response, provide the formal definition, explicitly list the three types of elements that can be combined to form an expression, and clarify whether a single number (like a fixed $10 processing fee) or a single variable (like 'n' for number of shipments) is considered an expression on its own.

Defining Algebraic Expressions for Professional Guidelines

A benefits specialist is documenting a formula used in a longevity bonus calculator. The entry `Years_of_Service * 500` is used to determine a payout. According to algebraic definitions, identify the correct term for this entry and list the three specific types of mathematical components it contains.

Benefit Calculation Definitions

A number, a variable, or numbers and variables joined by operation symbols is called an algebraic _____.

Which best describes the algebraic expression $$p - 10$$?

Put the parts in order to form the definition of an **algebraic expression**.

$$q + 50$$ is an algebraic expression.

Is the single number $$7$$ an algebraic expression?

Question:
What is $$n + 15$$ called, and what three kinds of parts can be combined to form it?

Sample answer:
It is an expression. The three parts are numbers, variables, and operation symbols.

Key points:
- $$n + 15$$ is an expression.
- Numbers are one part (for example, $$15$$).
- Variables are another part (for example, $$n$$).
- Operation symbols are the third part (for example, $$+$$).

Rubric:
Full credit: names it an 'expression' and lists all three parts (numbers, variables, operation symbols). Partial credit: correct name or a partial list. No credit: wrong name and no correct parts.

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