Foundations of Algebra

Simplifying Algebraic Expressions

Using the distributive property and combining like terms.

Foundational20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

The Real Number System

Two Tools for Simplifying

Simplifying an algebraic expression means rewriting it in an equivalent but shorter form, with no operations left that can be carried out. Two tools handle almost every simplification you'll do in this course: the distributive property, which clears parentheses, and combining like terms, which merges terms that share the same variable part.

Definition — The Distributive Property

For any numbers a, b, and c: a(b + c) = ab + ac. Multiplying a sum by a number is the same as multiplying each addend separately and then adding the results.

Worked Example — Distributing Across a Parenthesis

Simplify 3(2x + 5). Distribute the 3 across both terms inside the parentheses: 3(2x) + 3(5) = 6x + 15. There is nothing left to combine, so 6x + 15 is fully simplified.

Worked Example — Distributing a Negative

Simplify −2(4x − 3). Distribute −2 across both terms, carefully tracking signs: (−2)(4x) + (−2)(−3) = −8x + 6. A negative times a negative gives a positive, which is why the −3 becomes +6.

Once every parenthesis is cleared, the next step is combining like terms — adding or subtracting terms that share the same variable raised to the same power, exactly as with numerical coefficients.

Worked Example — Distributing, Then Combining

Simplify 4(x + 2) − 3x. First distribute: 4x + 8 − 3x. Now combine the like terms, 4x and −3x: (4 − 3)x + 8 = x + 8.

4(x + 2) - 3x = 4x + 8 - 3x = x + 8

(1)

Equation Editor

Edit as raw LaTeX

sum(i, 1, n, i^2) = n(n+1)(2n+1)/6

Constants

Structures

Calculus & discrete math

Greek

\sum_{i=1}^{n} i^{2} = \frac{\operatorname{n}\left(n + 1\right) \cdot \left(2 \cdot n + 1\right)}{6}

Evaluate

Unknown function "n"

Tip

Work in a fixed order: distribute first to eliminate every parenthesis, then combine like terms. Trying to combine terms while a parenthesis is still in the way is one of the most common sources of sign errors.

Common Mistakes

Distributing only to the first term inside the parentheses, such as simplifying 3(2x + 5) as 6x + 5.
The number outside the parentheses multiplies every term inside, not just the first one. Distribute to each term separately before moving on.
Losing track of a negative sign when distributing, such as simplifying −2(4x − 3) as −8x − 6.
Multiply signs carefully term by term: −2 times −3 is positive 6, not negative. Rewriting subtraction as 'adding a negative' before distributing can help keep the signs straight.

Key Takeaways

The distributive property, a(b + c) = ab + ac, clears a parenthesis by multiplying every term inside it.
Combine like terms only after every parenthesis has been cleared.
Track each term's sign carefully, especially when distributing a negative number.

Summary

Distributing and combining like terms are the two moves you'll use to simplify almost every expression in this course — and they're exactly what you'll need next to isolate a variable in a multi-step equation.

Track your progress

← The Real Number System Back to Foundations of Algebra