Expand algebraic expressions using the distributive law: a(b + c) = ab + ac and (a+b)(c+d) = ac + ad + bc + bd. Get detailed work, avoid common mistakes, and visualize with area models.
The distributive property represents the total area of a rectangle split into smaller parts: width × (length₁ + length₂) = width×length₁ + width×length₂.
The distributive property is fundamental in algebra for simplifying expressions, solving equations, and factoring. It's the algebraic version of the geometric idea that the area of a whole rectangle equals the sum of areas of its parts.
The distributive property states that multiplying a sum by a number gives the same result as multiplying each addend individually and then summing the products. Formally: a × (b + c) = a·b + a·c. This rule extends to subtraction: a(b − c) = ab − ac. It is one of the core axioms of arithmetic and algebra and appears in nearly every branch of mathematics, from elementary operations to polynomial manipulation and matrix algebra.
? General Form: a(b + c + d + ...) = ab + ac + ad + ...
? For binomial multiplication: (a+b)(c+d) = a(c+d) + b(c+d) = ac + ad + bc + bd (FOIL)
Our calculator provides a detailed breakdown for each expansion. In Basic Mode, we multiply the coefficient by every term inside parentheses, showing intermediate simplifications. In Binomial Mode, we apply the FOIL method (First, Outer, Inner, Last) and combine like terms. This process mirrors how students learn to expand expressions in pre-algebra and algebra I courses.
The internal parser handles rational numbers, decimals, negative signs, and variables. For basic mode, the inner expression is tokenized into terms (each term has a coefficient and an optional variable part). Each term is then multiplied by the outer coefficient, and the final expression is rebuilt with like‑term grouping. Binomial mode uses linear binomial parsing (ax + b format) and performs full FOIL expansion, followed by simplification of quadratic, linear, and constant terms. All steps are displayed transparently, ensuring trust and understanding.
Note on binomials: Both binomials must contain the same variable (e.g., x). Mixed-variable input (like (x+2)(y+3)) is not supported and will trigger an error message. We recommend using our basic mode for multi‑variable distributions.
Using FOIL: First: 3x·x = 3x²; Outer: 3x·(-4) = -12x; Inner: 2·x = 2x; Last: 2·(-4) = -8 → Combine -12x+2x = -10x → Result: 3x² -10x -8. This matches our calculator's output and step breakdown, reinforcing conceptual learning.
| Expression | Expanded Form | Explanation |
|---|---|---|
| 4(x + 3) | 4x + 12 | 4·x + 4·3 |
| -2(3y - 5) | -6y + 10 | -2·3y + (-2)·(-5) |
| (x+2)(x+5) | x² + 7x + 10 | x·x + x·5 + 2·x + 2·5 |
| (2a-3)(a+4) | 2a² + 5a - 12 | FOIL and combine like terms |
| ½(4z + 6) | 2z + 3 | ½·4z = 2z; ½·6 = 3 |
The distributive property has been implicitly used since ancient Babylonian mathematics (c. 1800 BCE) for area calculations. Euclid's "Elements" (300 BCE) contains geometric proofs of the distributive law (Book II). The formal algebraic statement emerged with the development of symbolic algebra in the 16th-17th centuries. Today, it stands as a fundamental property of fields and rings in abstract algebra, connecting addition and multiplication.