Distribution Patterns

Algebra 1 · Axiom Academy

From distributing a single term to FOIL to multiplying any two polynomials — it is always the same rule: every term meets every term. Start with the simplest case: a single term multiplying a sum. The term reaches in and multiplies each part inside the parentheses — the parentheses are not a barrier, they are a list of things to hit. The 2x must reach BOTH terms inside 2. Two Binomials: the FOIL Pattern Now both factors are sums. Each term in the first binomial must reach each term in the second — that is four products. FOIL is just a way to name the four arcs so none is missed: Watch the same four arcs on real numbers, then collect like terms. Here the Outer and Inner products are both multiples of x , so they combine into the middle term. FOIL only named four arcs because both factors had two terms. The real rule has no limit: a 2-term factor times a 3-term factor draws a full 2 × 3 grid of products — six of them — which we then collect. 5. The One Pattern Behind All Three Monomial distribution, FOIL, and the general case differ only in how many terms each factor has. Stack them up and the same grid just grows: , then , then cells. Every term of the first expression multiplies every term of the second; then collect like terms. You have seen distribution, FOIL, and general polynomial multiplication as one idea — every term of one factor meeting every term of the other. Scroll up to revisit any step.

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