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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 $$b^5$$.

Algebraic Term

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

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.

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