Box Method Calculator — Polynomial Multiplication
Enter the four coefficients of two linear binomials — a, b for (ax + b) and c, d for (cx + d) — and see the expanded polynomial alongside a full box-method breakdown that shows every step.
Set up: multiply each pair of terms
Top-left cell — outer × outer
Top-right cell — outer × inner
Bottom-left cell — inner × outer
Bottom-right cell — inner × inner
Combine the two like x-terms from the off-diagonal cells
Write out the final polynomial
How does this calculator work?
(ax + b)(cx + d) = ac·x² + (ad + bc)·x + bd. The box method places each term as a row and column header, fills four cells with products, then combines the two x-terms. Enter a, b, c, d to expand the binomials and see every multiplication step.
Formula
How this is calculated
The box (grid) method organises polynomial multiplication by placing one factor along the top of a 2×2 grid and the other along the side. Each cell is filled by multiplying its row header by its column header. For two binomials this gives four cells: ac·x² (outer × outer), ad·x (outer × inner), bc·x (inner × outer) and bd (inner × inner). Adding all four terms and collecting the two x-terms gives the standard trinomial form.
The method is equivalent to FOIL (First, Outer, Inner, Last) but the grid layout makes it much easier to avoid sign errors, especially when coefficients are negative. Because every multiplication step is isolated in its own cell, you can check each product independently before combining.
This calculator supports any real-number coefficients, including negatives and decimals. The result is always a quadratic trinomial (degree-2 polynomial) when both a and c are non-zero; if one of them is zero the result reduces to a lower-degree expression. The box method extends naturally to larger polynomials (trinomials, etc.) by adding more rows and columns.
Frequently asked questions
The box method (also called the grid method) is a visual way to multiply polynomials. You draw a rectangle divided into cells — one row or column per term of each factor — fill each cell with the product of its row and column headers, then add all the cells together and collect like terms.
For two binomials they give the same answer. FOIL is a mnemonic (First, Outer, Inner, Last) for the four products of exactly two binomials. The box method is a more general grid that works for polynomials with any number of terms.
Enter the negative value directly (e.g. d = −4 for x − 4). The calculator handles the signs automatically in each cell and in the final polynomial.
Also known as
TG we-Calculate Editorial Team. (2026). Box Method Calculator — Polynomial Multiplication [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/box-method-calculator
TG we-Calculate Editorial Team. "Box Method Calculator — Polynomial Multiplication." TG we-Calculate. 2026. https://we-calculate.com/calculator/box-method-calculator.
TG we-Calculate Editorial Team, "Box Method Calculator — Polynomial Multiplication," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/box-method-calculator
@misc{wecalculate_box_method_calculator, title = {Box Method Calculator — Polynomial Multiplication}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/box-method-calculator}}, year = {2026}, note = {TG we-Calculate} }
Did this calculator help you?