1Cademy - The Vertical Method for Multiplying Binomials

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Multiplying a Binomial by a Binomial Using the Distributive Property

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The Vertical Method for Multiplying Binomials

The Vertical Method is a technique for multiplying two polynomials that mirrors the familiar procedure for multiplying multi-digit whole numbers by hand. To use this method, write one polynomial above the other, just as you would stack two numbers for long multiplication. Multiply the top polynomial by each term of the bottom polynomial one at a time, writing each result (called a partial product) on its own line and aligning like terms in the same column. After all partial products have been computed, add them together by combining like terms column by column to obtain the final product.

For example, to multiply two-digit numbers like $23 \times 46$ , you first multiply $23$ by $6$ to get the partial product $138$ , then multiply $23$ by $4$ (tens digit) and line up the result in the correct columns, and finally add the partial products to get $1{,}058$ . The Vertical Method applies this same column-alignment strategy to polynomial expressions.

Unlike the FOIL method — which works only for multiplying two binomials — the Vertical Method works for multiplying any two polynomials, regardless of how many terms each one contains. Along with the Distributive Property and the FOIL method, it is one of three standard techniques for multiplying two binomials. The partial products generated by the Vertical Method are the same terms that appear in the FOIL method, just organized in a columnar layout rather than a horizontal sequence.

Updated 2026-04-29

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