The Distributive Property

The distributive property says a number outside parentheses multiplies everything inside. 3(x + 4) = 3x + 12.

Every term inside gets multiplied, not just the first one.

The dangerous case is a negative outside. −2(x − 5) means −2 times x and −2 times −5, giving −2x + 10. The last sign flips because negative times negative is positive.

A bare minus sign in front of parentheses is a −1: −(x − 7) = −x + 7. Every sign inside flips.

Distributivity of multiplication over addition, a(b + c) = ab + ac, is one of the field axioms — it is not derived from anything simpler, and nearly every algebraic manipulation depends on it.

It runs in both directions. Read left to right it expands; read right to left it factors, which is the entire basis of factoring later. Combining like terms, factoring out a GCF, multiplying binomials with FOIL, and polynomial multiplication in general are all this one axiom applied repeatedly.

For subtraction, use a(b − c) = ab − ac, which follows by writing b − c = b + (−c). The frequent error is treating the leading minus as a decoration rather than a factor of −1; writing −(3x − 5) explicitly as (−1)(3x − 5) makes the sign changes mechanical rather than remembered.