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Multiplying Polynomial by Polynomial

Multiplying a polynomial by a polynomial means multiplying every term in the first by every term in the second, then combining like terms. For two binomials, this is the FOIL method: First, Outer, Inner, Last. Learn the steps with worked examples, including multiplying beyond two binomials.

Multiplying a polynomial by a polynomial

To multiply polynomial by polynomial, multiply every term in the first polynomial by every term in the second, then combine like terms. For two binomials, this pairwise multiplication has a name: the FOIL method — First, Outer, Inner, Last, standing for the four term-pairs you multiply.

FOIL: First, Outer, Inner, Last — the four products when multiplying two binomials.

The FOIL method, step by step

For (x + 2)(x + 3): multiply the First terms (x × x = x²), the Outer terms (x × 3 = 3x), the Inner terms (2 × x = 2x), and the Last terms (2 × 3 = 6). Add all four products and combine like terms: x² + 3x + 2x + 6 = x² + 5x + 6.

Multiplying larger polynomials

FOIL only works for two binomials. For anything larger — a binomial times a trinomial, for example — the same idea still applies: multiply every term in the first factor by every term in the second, then combine like terms. This builds directly on multiplying a monomial by a binomial and multiplying binomial by binomial, just with more terms to track.

Example

(2x + 1)(x² + 3x + 4): multiply 2x by each term of the trinomial (2x³ + 6x² + 8x), then multiply 1 by each term (x² + 3x + 4), and add: 2x³ + 6x² + 8x + x² + 3x + 4 = 2x³ + 7x² + 11x + 4.

Related lessons

Algebra