Box Method Calculator (multiply polynomials with a grid / generic rectangle)

Understanding the box method polynomial multiplier — concept and principle

This calculator multiplies two polynomials with the box method, also called the generic rectangle or area model. It lays out the grid of partial products, collects like terms along the diagonals and gives the simplified product. Use it to learn or check polynomial multiplication.

How it is calculated

1. Write the terms of the first polynomial down the side (rows) and the terms of the second across the top (columns).
2. Each box = row term × column term: multiply the coefficients and add the exponents.
3. Boxes with the same power of x lie on the same diagonal; add them.
4. Write the result from the highest power down.

• A binomial times a binomial gives a 2 × 2 box (the same four products as FOIL); a binomial times a trinomial gives 2 × 3.

The principle

The box is an area model of the distributive property. If a rectangle has side lengths (a + b) and (c + d + e), its area is the sum of the smaller rectangles a·c, a·d, … , b·e. Every term of one polynomial meets every term of the other exactly once, so nothing is missed or doubled. Exponents add because xᵐ × xⁿ = xᵐ⁺ⁿ. Arranging the columns in descending powers puts like terms on diagonals, which makes collecting them easy.

Worked example

Default inputs: (2x + 3)(x² − 4x + 5).

1. Box size 2 × 3, 6 boxes; column headings x² │ −4x │ 5.
2. Row 2x: 2x³ │ −8x² │ 10x.
3. Row 3: 3x² │ −12x │ 15.
4. Diagonals: 2x³; −8x² + 3x² = −5x²; 10x − 12x = −2x; 15.
5. Product: 2x³ − 5x² − 2x + 15.

How to read the result

The degree of the product is the sum of the two degrees, here 1 + 2 = 3, and the number of boxes is the product of the numbers of terms. A quick check is to put x = 1 into both sides: (2 + 3)(1 − 4 + 5) = 5 × 2 = 10, and 2 − 5 − 2 + 15 = 10.

Common mistakes

• Multiplying exponents instead of adding them.
• Losing a negative sign when copying a term into the heading.
• Combining boxes that are not like terms.
• Forgetting a box when one polynomial has a missing power.

Frequently asked questions

What is the box method in algebra?

A grid method for multiplying polynomials: each box holds the product of one term from each polynomial.

Is the box method the same as FOIL?

For two binomials it gives the same four products; it also works for longer polynomials.

How do I check a polynomial product?

Substitute a simple value such as x = 1 or x = 2 into both the factors and the product.

Sources: OpenStax, Elementary Algebra 2e (2020), §6.3 Multiply Polynomials · OpenStax, Algebra 1 (2025), area model for polynomials