What Is the Box Method?
The box method (also called the area model or grid method) is a visual technique for multiplying polynomials. Instead of relying on the traditional FOIL method, which only works for multiplying two binomials, the box method organizes all partial products into a rectangular grid. This makes it easier to keep track of every term, especially when multiplying polynomials with more than two terms.
The method gets its name from the rectangular grid (or "box") that is drawn to organize the multiplication. Each term of the first polynomial labels a row, and each term of the second polynomial labels a column. The cells of the grid contain the products of the corresponding row and column terms.
How the Box Method Works
- Set up the grid: Draw a rectangle divided into cells. Place the terms of the first polynomial along the left side and the terms of the second polynomial along the top.
- Fill in partial products: Multiply each row term by each column term and write the product in the corresponding cell.
- Combine like terms: Identify all terms with the same degree (power of x) and add their coefficients together.
- Write the final answer: Arrange the combined terms in descending order of degree.
Examples
Binomial x Binomial
Multiply (2x + 3)(x + 4) using a 2x2 grid. The four cells give partial products that combine into the final trinomial.
Binomial x Trinomial
Multiply (x + 2)(x^2 - 3x + 1) using a 2x3 grid. Six partial products combine into a cubic polynomial.
Perfect Square
Multiply (a + b)^2 = (a + b)(a + b) using the box method to visualize a^2 + 2ab + b^2.
Box Method vs. FOIL
FOIL (First, Outer, Inner, Last) is a mnemonic specifically for multiplying two binomials. While it works well for that specific case, it breaks down when dealing with trinomials or higher-degree polynomials. The box method, on the other hand, scales naturally to any number of terms. A binomial times a trinomial uses a 2x3 grid, a trinomial times a trinomial uses a 3x3 grid, and so on.
Advantages of the Box Method
- Works for polynomials of any size, not just binomials.
- Provides a visual and organized layout that reduces errors.
- Makes it easy to identify and combine like terms by looking at diagonal patterns in the grid.
- Helps students understand the distributive property in a concrete way.
- Can also be used in reverse for factoring polynomials.
Using the Box Method for Factoring
The box method can also be used in reverse to factor polynomials. If you know the product and one factor, you can set up the grid and work backward to find the other factor. This is particularly useful for factoring trinomials like ax2 + bx + c where a is not equal to 1.
Connection to the Area Model
The box method is closely related to the area model used in elementary arithmetic. Just as you can visualize 23 x 14 as (20 + 3)(10 + 4) using a grid of areas, you can visualize polynomial multiplication the same way. Each cell of the grid represents a "partial area" (or partial product) that contributes to the total.
Tips for Using the Box Method
- Always write polynomials in descending order of degree before setting up the grid.
- Include terms with zero coefficients as placeholders (e.g., 0x) to keep the grid organized.
- Look for diagonal patterns to find like terms quickly.
- Double-check by substituting a simple value (like x = 1) into both the original expression and your answer.