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\(a\)
\(x\)
\(+\)
\(b\)

Terms, Factors, &
Coefficients

Learn to break apart algebraic expressions and identify the building blocks that give every formula its meaning.

Algebra 2

What Is an Algebraic Expression?

An algebraic expression is a combination of numbers, variables, and operations: no equals sign, no inequality. It represents a quantity that can change depending on the values of its variables.

Every expression can be decomposed into terms, and every term can be decomposed further into factors and coefficients. Mastering this skill is the first step to simplifying, factoring, and solving.

Defining a Term

A term is a single piece of an expression that is separated from other pieces by a + or - sign. Each term is a product of numbers and variables.

\(3x^2 + 5x - 7\)
 Term 1
\(3x^2\)
 Term 2
\(5x\)
 Term 3
\(-7\)

What Are Factors?

Factors are the quantities being multiplied together within a single term. When you write \(3x^2\), the factors are \(3\), \(x\), and \(x\) (or equivalently \(3\) and \(x^2\)).

\(3x^2\)
\(3\)
 Numerical factor
\(x^2\)
 Variable factor
\(3 \times x^2 = 3x^2\)

What Is a Coefficient?

A coefficient is the numerical factor that multiplies the variable part of a term. It tells you how many or how much of the variable quantity you have.

\(7x^3\)
Coefficient: 7
\(-4y\)
Coefficient: -4
\(x^2\)
Coefficient: 1 (implied)

The Constant Term

A constant term is a term with no variable: just a number on its own. It does not change when the variable changes.

\(2x^2 - 9x + 12\)
\( \text{Constant} = 12 \)

Formal Definitions

1
Term: A single number, variable, or the product of numbers and variables separated by \(+\) or \(-\) signs.
2
Factor: Any quantity that is multiplied to form a term. In \(6ab\), the factors are \(6\), \(a\), and \(b\).
3
Coefficient: The numerical factor of a term. If no number is written, it is \(1\) (or \(-1\) if only a minus sign appears).
4
Constant: A term that contains no variables. Its value never changes.

Anatomy of an Expression

\(-2x^3 + 5x - 8\)
\(-2x^3\)
\(-2\)
 coefficient
\(x^3\)
 variable factor
\(5x\)
\(5\)
 coefficient
\(x\)
 variable factor
\(-8\)
 constant term (no variable factor) Coefficient Variable Factor Constant

Terms vs. Factors: The Key Difference

Terms
Separated by addition or subtraction. Each term stands on its own.
Factors
Connected by multiplication. They live inside a single term.

In \(4xy + 9z\), the expression has 2 terms. The first term, \(4xy\), has three factors: \(4\), \(x\), and \(y\).

How to Decompose Any Expression

1
Identify the terms. Look for \(+\) and \(-\) signs that split the expression.
2
Count the terms. Each piece separated by \(+\) or \(-\) is one term (keep the sign with the term).
3
Find the coefficient. For each term, locate the numerical part multiplying the variables.
4
List the factors. Break each term into everything being multiplied together.
5
Spot constants. Any term without a variable is a constant term.

Example 1: \(6a^2 - 3ab + 10\)

\(6a^2 - 3ab + 10\)
\(6a^2\)
\( \text{coeff}: 6 \)
\( \text{factors}: 6, a^2 \)
\(-3ab\)
\( \text{coeff}: -3 \)
\( \text{factors}: -3, a, b \)
\(10\)
 constant
\( \text{coeff}: 10 \)

Example 2: \(\dfrac{m}{4} - n^2 + 2mn\)

This expression has three terms. Notice the first term has a fraction as its coefficient:

\(\frac{m}{4}\)
Coefficient: \(\frac{1}{4}\)
Variable: \(m\)
\(-n^2\)
Coefficient: \(-1\)
Variable: \(n^2\)
\(2mn\)
Coefficient: \(2\)
Factors: \(2, m, n\)

Example 3: \(x(2x + 3) - 5\)

Before identifying parts, expand the expression if terms are grouped inside parentheses:

1
Distribute: \(x(2x + 3) = 2x^2 + 3x\)
2
Rewrite: \(2x^2 + 3x - 5\)
3
Identify: Three terms with coefficients \(2\), \(3\), and constant \(-5\).

Common Mistakes

✕ Dropping the Sign
Saying the coefficient of \(-7x\) is \(7\) instead of \(-7\). The sign belongs to the term.
✕ Invisible Coefficient
Forgetting that \(x\) has coefficient \(1\) and \(-y^3\) has coefficient \(-1\).
✕ Mixing Terms and Factors
Saying \(x\) and \(5\) are terms in \(5x\). They are factors: the expression \(5x\) is a single term.

Summary at a Glance

Concept Definition Example in \(4x^2 - 3x + 7\)
Term Part separated by \(+\) or \(-\) \(4x^2\), \(-3x\), \(7\)
Factor Quantities multiplied within a term \(4\) and \(x^2\) are factors of \(4x^2\)
Coefficient Numerical factor of a term \(4\), \(-3\), \(7\)
Constant Term with no variable \(7\)

Question 1 of 5

How many terms does the expression \(9a^3 - 2a + 4b - 11\) have?

Question 2 of 5

What is the coefficient of the term \(-8xy^2\)?

Question 3 of 5

Which of the following lists all factors of the term \(12p^2q\)?

Question 4 of 5

In the expression \(k - 3k^4 + 2\), what is the coefficient of the first term?

Question 5 of 5

Which statement about the expression \(5x^2 + 3x - 9\) is true?