In short
An algebraic expression is a combination of numbers, variables (letters standing for unknown or arbitrary values), and operations (+, -, \times, \div, powers). Each chunk separated by + or - signs is a term; the number in front of the variable part is the coefficient; the sum of the exponents on the variables in a term is the degree of that term. Two terms are like terms if they share exactly the same variable part — and only like terms can be combined by adding or subtracting. The operations on expressions — addition, subtraction, multiplication — follow the same rules as on numbers (commutative, associative, distributive), and simplifying an expression means applying those rules until no more like terms remain.
Think about the perimeter of a rectangle. If the length is l and the width is w, the perimeter is 2l + 2w. That expression is not a number — it is a recipe that produces a number once you plug in specific values for l and w. If the length is 5 and the width is 3, the perimeter is 2(5) + 2(3) = 16. If the length is 12 and the width is 7, the perimeter is 2(12) + 2(7) = 38.
The recipe 2l + 2w is an algebraic expression. It carries the structure of the calculation — which operations to apply in which order — without committing to any particular numbers. This is the entire point of algebra: write the structure once, reuse it for any numbers you need.
Every equation you will ever solve, every function you will ever graph, every formula you will ever use in physics or chemistry or economics, is built from algebraic expressions. This article is the vocabulary lesson: what the parts are called, how to tell which parts can combine and which cannot, and how to perform the four operations on expressions themselves.
Variables and constants
A variable is a letter that stands for a number whose value is either unknown or deliberately left unspecified. In 3x + 7, the letter x is a variable. It could be 2, or -5, or \pi, or any real number — the expression has a different value for each choice.
A constant is a fixed number. In 3x + 7, the 3 and the 7 are constants. So is \pi when it appears in 2\pi r — even though \pi is a letter, its value is fixed (approximately 3.14159), so it acts as a constant.
The distinction matters because variables are what make an expression flexible. The expression 3(5) + 7 = 22 is arithmetic — a single computation with a single answer. The expression 3x + 7 is algebra — a family of computations, one for each value of x.
Terms, coefficients, and degree
An algebraic expression is a sum of terms. Each term is a product of a numerical part (the coefficient) and a variable part (one or more variables raised to non-negative integer powers).
Here are the parts for several terms:
| Term | Coefficient | Variable part | Degree |
|---|---|---|---|
| 5x^3 | 5 | x^3 | 3 |
| -2xy | -2 | xy | 2 (1 + 1) |
| 7 | 7 | (none) | 0 |
| x | 1 | x | 1 |
| -y^2 | -1 | y^2 | 2 |
| 4x^2y^3 | 4 | x^2y^3 | 5 (2 + 3) |
The degree of a term is the sum of all the exponents on its variables. The term 4x^2y^3 has degree 2 + 3 = 5. A plain number like 7 has degree 0 (there are no variables, so the exponent sum is 0). A single variable x has degree 1 (since x = x^1).
The degree of an expression is the highest degree among all its terms. The expression 3x^2 + 5x - 1 has terms of degree 2, 1, and 0, so the expression has degree 2.
Two conventions catch people the first time:
- When a variable stands alone, like x, its coefficient is 1, not 0. You are multiplying by 1; you just don't write it.
- When a term has a minus sign, like -y^2, the coefficient is -1, not 1. The sign is part of the coefficient.
Like and unlike terms
Two terms are like terms if they have exactly the same variable part — the same variables, each raised to the same power. Only the coefficient can differ.
| Like terms | Unlike terms |
|---|---|
| 3x^2 and -7x^2 | 3x^2 and 3x^3 (different exponent) |
| 5xy and -xy | 5xy and 5x (second has no y) |
| 4 and -9 | 4 and 4x (second has a variable) |
Like terms can be combined by adding their coefficients: 3x^2 + (-7x^2) = (3 - 7)x^2 = -4x^2. This is the distributive law in reverse — you are factoring out the common variable part.
Unlike terms cannot be combined. 3x^2 + 5x stays as 3x^2 + 5x; there is no simpler form. Trying to add them is like trying to add 3 apples and 5 oranges — the units don't match, so you keep them separate.
Operations on algebraic expressions
All four operations work on expressions, and they all come from the same rules you met in Operations and Properties: commutativity, associativity, and distributivity. Here is each one.
Addition and subtraction
To add two expressions, collect all the terms together and combine like terms.
The first equals sign uses the associative and commutative laws (rearranging terms). The second uses the distributive law in reverse (combining like terms: 3 + 1 = 4 for x^2, 2 - 4 = -2 for x, -5 + 7 = 2 for the constants).
Subtraction is the same, but with every term in the second expression negated:
The minus sign in front of the bracket distributes to every term inside: -(x^2) = -x^2, -(-4x) = +4x, -(7) = -7. The most common mistake is forgetting to flip the sign on the -4x term — which becomes +4x, not -4x.
Multiplication
To multiply two expressions, use the distributive law repeatedly. Each term in the first expression multiplies every term in the second.
The first step is distributivity (four products, one for each pair of terms). The second step is combining like terms (-8x + 3x = -5x).
For multiplying single terms, use the laws of exponents from Exponents and Powers: multiply the coefficients, and add the exponents of each variable.
Division
Division of expressions by a single term distributes across the sum:
Each term is divided individually, using the quotient law of exponents (x^3 / x = x^2, etc.). Division by an expression with more than one term is a more involved process (polynomial long division), which you meet in Polynomials — Introduction.
Evaluating an expression
To evaluate an expression at a specific value of its variable, substitute the value everywhere the variable appears and compute.
Take f(x) = 2x^2 - 5x + 3 at x = 4:
Use brackets around the substituted value, especially when it is negative. At x = -1:
The bracket in 2(-1)^2 matters — it ensures the square applies to -1 and gives +1, not -(1^2) = -1.
The figure below lets you evaluate 2x^2 - 5x + 3 interactively. Drag the red point along the curve to see how the value changes.
Two worked examples
Example 1: Simplify $(4x^2 - 3x + 5) - 2(x^2 + x - 6)$
This requires distributing the -2 across the second expression and then collecting like terms.
Step 1. Distribute the -2 into the second bracket.
Why: the distributive law says -2 multiplies each term inside the bracket. Signs matter: -2 \times (-6) = +12. The bracket is now gone.
Step 2. Write all terms together.
Why: this is just the first expression followed by the result of Step 1. No brackets remain, and you can now see all six terms laid out.
Step 3. Group like terms by their variable part.
Why: rearranging by the commutative and associative laws puts like terms side by side — x^2 terms together, x terms together, constants together. This makes the combining step mechanical.
Step 4. Combine each group.
Result. (4x^2 - 3x + 5) - 2(x^2 + x - 6) = 2x^2 - 5x + 17.
A quick check: plug in x = 1. Original: (4 - 3 + 5) - 2(1 + 1 - 6) = 6 - 2(-4) = 6 + 8 = 14. Simplified: 2 - 5 + 17 = 14. They match.
Example 2: Multiply and simplify $(3a - 2b)(a + 4b)$
This is a product of two binomials in two variables. Each term in the first binomial multiplies each term in the second — four products in total.
Step 1. Apply the distributive law: each term of (3a - 2b) multiplies each term of (a + 4b).
Why: the distributive law extends to four products when each expression has two terms. This is sometimes called the FOIL method (First, Outer, Inner, Last), but the name is just a memory aid — the rule is distributivity.
Step 2. Compute each product.
Why: for each product, multiply the coefficients and use the exponent product law. 3a \cdot a = 3a^2. 3a \cdot 4b = 12ab. (-2b) \cdot a = -2ab. (-2b) \cdot 4b = -8b^2.
Step 3. Combine like terms.
The terms 12ab and -2ab are like terms (both have variable part ab). Combine: 12 - 2 = 10.
Why: 3a^2 and -8b^2 are unlike terms (different variable parts), so they stay. Only the ab terms combine.
Result. (3a - 2b)(a + 4b) = 3a^2 + 10ab - 8b^2.
Check: let a = 2, b = 1. Original: (6 - 2)(2 + 4) = 4 \times 6 = 24. Simplified: 3(4) + 10(2) - 8(1) = 12 + 20 - 8 = 24. They match.
Common confusions
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"3x + 5x^2 = 8x^3." No. 3x and 5x^2 are unlike terms — one is degree 1, the other degree 2. They cannot be combined. The expression 3x + 5x^2 is already fully simplified. A common variant of this error is writing x + x^2 = x^3, which confuses addition with multiplication.
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"The coefficient of x is 0." When x appears by itself, its coefficient is 1: x = 1 \cdot x. The coefficient is the number you are multiplying the variable by. If that number were 0, the term would vanish entirely (0 \cdot x = 0).
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"(a + b)^2 = a^2 + b^2." This is the most famous algebra mistake, and it is wrong. The correct expansion is (a + b)^2 = a^2 + 2ab + b^2. The cross-term 2ab comes from the distributive law: (a + b)(a + b) = a^2 + ab + ba + b^2 = a^2 + 2ab + b^2. You can verify with numbers: (3 + 4)^2 = 49, but 3^2 + 4^2 = 25. The missing 2(3)(4) = 24 is exactly the gap.
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"Distributing the minus sign only flips the first term." When you write -(x^2 - 3x + 2), the minus distributes to every term: -x^2 + 3x - 2. The sign of each term flips — not just the first. Forgetting this is the most common source of sign errors in simplification.
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"5xy and 5yx are unlike terms." They are the same term. Multiplication is commutative (xy = yx), so 5xy = 5yx. The order of the variables does not matter — only their identity and exponents.
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"Degree is the exponent of the last variable." No — degree is the sum of the exponents of all variables in a term. The term 4x^2y^3 has degree 2 + 3 = 5, not 3 and not 2.
Going deeper
If you came here for the vocabulary — terms, coefficients, degree, like terms — and the rules for adding, subtracting, and multiplying expressions, you have everything you need. What follows is for readers who want to see how algebraic expressions connect to more formal structures.
Expressions vs. equations vs. identities
An expression is a recipe that produces a value: 3x^2 + 2x - 5. An equation is a statement that two expressions are equal: 3x^2 + 2x - 5 = 0. Equations have solutions — specific values of x that make them true.
An identity is an equation that is true for all values of the variable, not just specific ones. The statement (a + b)^2 = a^2 + 2ab + b^2 is an identity — it holds for every a and every b. In contrast, x^2 - 1 = 0 is an equation with specific solutions (x = 1 and x = -1), not an identity.
The distinction matters because identities can be used as rewriting rules: any time you see (a + b)^2 in any context, you can replace it with a^2 + 2ab + b^2. Equations cannot be used this way — they are only true at their solutions.
Polynomials: expressions with a specific shape
A polynomial is an algebraic expression where every term has a non-negative integer exponent on the variable. So 3x^2 + 2x - 5 is a polynomial, but 3x^{-1} + 2\sqrt{x} is not (the first term has a negative exponent, the second has a fractional one). Polynomials are the cleanest, best-behaved family of expressions, and they get their own detailed article in Polynomials — Introduction.
Why "like terms" is the distributive law
The rule "combine like terms by adding coefficients" is just the distributive law written backwards. When you write 3x^2 + 5x^2 = 8x^2, you are really writing 3 \cdot x^2 + 5 \cdot x^2 = (3 + 5) \cdot x^2 = 8x^2 — you factored out the common factor x^2. The "like terms" language is a convenient shortcut, but behind it there is only one rule: distributivity.
This is why unlike terms cannot be combined. The expression 3x^2 + 5x has no common variable factor to pull out (one term has x^2, the other has x). There is no single rule that lets you merge them into one term. They are genuinely different animals.
Expressions in multiple variables
Everything on this page extends to expressions in two, three, or more variables. The expression 3x^2y + 2xy^2 - 5xyz has three terms. Like terms require all variables to match: 3x^2y and 7x^2y are like terms, but 3x^2y and 3xy^2 are not. The degree of each term is the sum of all the exponents: 3x^2y has degree 3, 2xy^2 has degree 3, and 5xyz has degree 3 — so this is a homogeneous polynomial of degree 3.
Multi-variable expressions appear everywhere in geometry (the equation of a circle is x^2 + y^2 = r^2), in physics (kinetic energy is \frac{1}{2}mv^2), and in economics (revenue is price times quantity, R = pq). The vocabulary — terms, coefficients, degree, like terms — is the same regardless of how many variables are in play.
Where this leads next
Algebraic expressions are the raw material of everything that follows in algebra.
- Operations and Properties — the foundational rules (distributive, commutative, associative) that govern every operation on expressions.
- Fractions and Decimals — where the same fraction rules apply to rational expressions (fractions whose numerators and denominators are polynomials).
- Polynomials — Introduction — the systematic study of polynomial expressions: degree, roots, factoring, and long division.
- Exponents and Powers — the exponent laws that power the multiplication and division of single terms.
- Laws of Exponents in Algebra — where the exponent laws are applied to algebraic expressions rather than individual numbers.