Distributive Property Calculator

What is a Distributive Property Calculator?

A Distributive Property Calculator applies the identity \(a(b+c)=ab+ac\) to algebraic expressions so you can expand, simplify, or factor them quickly and reliably. It handles integers, fractions, decimals, and symbols; respects order of operations; and keeps track of signs to prevent common mistakes. Whether you need to distribute a single term across a sum/difference, expand products like \((p+q)(r+s)\), or reverse the process by factoring out a greatest common factor (GCF), the calculator shows the intermediate lines, then combines like terms into a clean final form suitable for homework, exams, and programming tasks.

About the Distributive Property Calculator

The tool parses your input, normalizes it to standard form, and applies distributive steps systematically. When expanding, it multiplies each term in a parenthesis by the outside factor, accumulates results, and merges like terms. For binomial products, it executes the distributive law twice (often remembered as FOIL), producing all pairwise products. For factoring, it searches for a nontrivial GCF among coefficients and variables, then rewrites the sum or difference as a product of the GCF and a simplified parenthesis. It also supports sign-sensitive cases such as distributing a negative or factoring out a negative to make leading coefficients positive. Exact rational arithmetic is used where possible so results are precise, not rounded.

How to Use this Distributive Property Calculator

1. Enter an expression to expand (e.g., 3(x+4), (x+2)(x-3)) or a sum to factor (e.g., 8x+12).
2. Select the mode: Expand (apply distribution) or Factor (extract GCF using distribution in reverse).
3. Press calculate to see step-by-step distribution and the final simplified expression with like terms combined.
4. Review each line to understand sign handling, coefficient multiplication, and variable exponent rules.
5. Copy the final expression for your worksheet, code, or documentation.

Core Formulas (LaTeX)

Distributive law over addition: \[ a(b+c)=ab+ac. \]

Distributive law over subtraction: \[ a(b-c)=ab-ac. \]

Generalized (sum of many terms): \[ a\!\left(\sum_{i=1}^{n} t_i\right)=\sum_{i=1}^{n} (a\,t_i). \]

Binomial–binomial (double distribution): \[ (p+q)(r+s)=pr+ps+qr+qs. \]

Factoring out the GCF (reverse distribution): \[ ab+ac=a(b+c). \]

Examples (Illustrative)

Example 1 — Simple expansion

\(3(x+4)=3x+12\).

Example 2 — Expansion with a negative

\(-2(5y-3)=-10y+6\).

Example 3 — Binomial product via distribution

\((x+2)(x-3)=x(x-3)+2(x-3)=x^2-3x+2x-6=x^2-x-6\).

Example 4 — Factoring by GCF

\(8x+12=4(2x+3)\).

Example 5 — Fractions

\(\tfrac{1}{2}(4a-6b)=2a-3b\).

FAQs

What is the distributive property?

It’s the identity \(a(b+c)=ab+ac\), letting you multiply a term across a sum or difference.

What’s the difference between expanding and factoring?

Expanding applies distribution to remove parentheses; factoring reverses it to extract a common factor.

Does the calculator handle negatives correctly?

Yes. It distributes signs to every term and can factor out a negative for cleaner leading terms.

Can it combine like terms automatically?

Yes. After distribution, coefficients of identical variable powers are merged into a simplified result.

Will it work with fractions and decimals?

Absolutely. It multiplies coefficients exactly for rationals and precisely for decimals.

Is FOIL different from distribution?

FOIL is just double distribution for two-term parentheses; it’s the same rule applied twice.