Multiplying polynomials without losing a term

Distribution scales to any number of terms

For each term in the first polynomial, distribute across the entire second polynomial. A grid or vertical layout can make the pairings visible.

FOIL is only the two-by-two case. Relying on the acronym becomes fragile as soon as a trinomial appears.

Multiply coefficients and add exponents

Within each product, multiply the numerical coefficients and use the product rule for matching variable bases.

Carry signs with their coefficients. Writing negative terms explicitly reduces accidental sign changes.

Combine only after all products exist

Premature combining makes it easy to omit a pairing. List the products first, then group identical powers.

A degree check helps: the product degree should usually equal the sum of factor degrees when leading coefficients are nonzero.

Worked example

Common mistakes

• Multiplying only the first and last terms.
• Adding rather than multiplying coefficients.
• Combining unlike powers of x.

Keep these ideas

• Every term pairs with every term.
• Signs belong to coefficients.
• Combine like terms at the end.