Expand Calculator — Expand and Simplify Brackets Free
Our expand calculator instantly expands brackets and simplifies algebraic expressions with step-by-step working. Whether you need to expand single brackets like 3(x + 4), expand double brackets like (x + 2)(x - 5), or tackle polynomial expansion with higher powers, this tool shows every step of the expansion process so you can learn while you calculate.
This expanding brackets calculator handles all types of bracket expansion: the distributive property for single brackets, the FOIL method for double brackets, and systematic multiplication for triple brackets and beyond. Enter any algebraic expression containing brackets, and the calculator will expand, combine like terms, and present the fully simplified result.
How to Expand and Simplify Expressions
To use this expand calculator, type your expression into the input field using standard algebraic notation. Use parentheses for brackets, the ^ symbol for powers, and * for multiplication (though implied multiplication like 2x works automatically). Press "Expand" or hit Enter to see the expanded and simplified result.
The calculator applies the distributive property systematically to remove all brackets. For an expression like (2x + 3)(x - 4), it multiplies each term in the first bracket by each term in the second bracket: 2x × x + 2x × (-4) + 3 × x + 3 × (-4) = 2x² - 8x + 3x - 12, then combines like terms to give 2x² - 5x - 12.
Expanding Single Brackets — Distributive Property
Expanding a single bracket uses the distributive property: multiply the term outside the bracket by every term inside. This is the foundation of all bracket expansion and the first expansion technique students learn in algebra.
Worked examples:
- 3(x + 4) = 3 × x + 3 × 4 = 3x + 12
- -2(x - 5) = -2 × x + (-2) × (-5) = -2x + 10
- 5(3a + 2b - c) = 15a + 10b - 5c
- x(x + 7) = x × x + x × 7 = x² + 7x
Our distributive property calculator provides additional focused practice on this fundamental expansion rule with detailed explanations.
Expanding Double Brackets — The FOIL Method
When expanding double brackets (two sets of parentheses multiplied together), use the FOIL method: multiply the First terms, Outer terms, Inner terms, and Last terms. This expanding brackets technique ensures you multiply every term in the first bracket by every term in the second bracket.
Example: Expand (x + 3)(x + 5)
- First: x × x = x²
- Outer: x × 5 = 5x
- Inner: 3 × x = 3x
- Last: 3 × 5 = 15
Combine like terms: x² + 5x + 3x + 15 = x² + 8x + 15
For step-by-step FOIL expansion with detailed guidance, try our dedicated FOIL calculator.
Expanding Triple Brackets and Polynomial Expansion
To expand triple brackets, first expand any two of the three brackets, then multiply that result by the remaining bracket. This polynomial expansion calculator handles this automatically, but understanding the process is valuable for exam preparation.
Example: Expand (x + 1)(x + 2)(x + 3)
- Step 1: Expand the first two brackets: (x + 1)(x + 2) = x² + 3x + 2
- Step 2: Multiply the result by the third bracket:
(x² + 3x + 2)(x + 3) = x³ + 3x² + 3x² + 9x + 2x + 6 - Step 3: Combine like terms: x³ + 6x² + 11x + 6
The expand calculator also handles polynomial powers like (x + 1)³ or (2x - 3)⁴ by applying repeated multiplication. For expansions involving trinomials like (x² + x + 1)², it multiplies each term systematically across all brackets and simplifies the result.
Perfect Squares and Difference of Squares
Certain bracket expansions follow special patterns that are worth recognizing. These algebraic expansion identities save time and appear frequently in exams and mathematical proofs.
- Perfect square (sum): (a + b)² = a² + 2ab + b² — e.g. (x + 4)² = x² + 8x + 16
- Perfect square (difference): (a - b)² = a² - 2ab + b² — e.g. (x - 3)² = x² - 6x + 9
- Difference of squares: (a + b)(a - b) = a² - b² — e.g. (x + 5)(x - 5) = x² - 25
- Cube of binomial: (a + b)³ = a³ + 3a²b + 3ab² + b³
To verify your expanded results by working backwards, use our factoring trinomials calculator to factor the expanded form back into brackets. Expanding and factoring are inverse operations — factoring undoes expansion.
Common Mistakes When Expanding Brackets
Even experienced algebra students make these expansion errors. Here are the most common pitfalls and how to avoid them:
- Forgetting to multiply all terms: In (x + 2)(x + 3), you must get four products (FOIL), not just x × x and 2 × 3.
- Sign errors with negatives: In -(x - 3), the result is -x + 3, not -x - 3. The negative distributes to every term.
- Squaring incorrectly: (x + 3)² is NOT x² + 9. You must expand: (x + 3)(x + 3) = x² + 6x + 9. Don't forget the middle term.
- Not combining like terms: After expanding, always check for terms that can be combined. For instance, x² + 5x + 3x + 15 simplifies to x² + 8x + 15.
About the Author
Marko Šinko
Co-Founder & Lead Developer, AI Math Calculator
When Do You Need to Expand and Simplify?
Expanding and simplifying brackets is essential in many areas of mathematics and beyond. Students encounter bracket expansion throughout their education, from GCSE and algebra classes through to university-level calculus and beyond. This expand and simplify calculator helps with:
- Solving equations: Many equations require expanding brackets before you can collect terms and solve for the unknown variable.
- Simplifying expressions: Expanding is often the first step in simplifying complex algebraic expressions by combining like terms.
- Calculus preparation: Before differentiating or integrating, polynomials often need to be expanded from factored form into a sum of terms.
- Checking factoring work: Expand your factored answer to verify it matches the original expression — expansion and simplification are how you confirm factoring is correct.
For related algebra tools, explore our combine like terms calculator for simplification practice, or our polynomial calculator for more advanced polynomial operations including addition, subtraction, and multiplication of polynomials.
