The box method for multiplying polynomials is basically the polynomial version of this cool trick that you can use to multiply regular numbers.
And it’s not the only way to multiply polynomials. There are three other methods that work just as well. They are:
The multiplication algorithm is my favorite method, but a lot of my students choose the box method as their favorite. So, it comes highly recommended 🙂
How to Use the Box Method to Multiply Polynomials
\[({\green {2x-8}})({\blue {3x+5}})={\red 6x^2}{\yellow-14x}{\red -40}\]
- Make a table and use the terms of the polynomials to label the columns and rows.
- Fill each cell of the table by multiplying the terms in the corresponding row and column.
- Combine like terms, if needed.
If there are any subtracted terms in the polynomials, write those as negative terms when you label the rows and columns.
Also, make sure to follow the rules for multiplying negative numbers when you multiply the coefficients of the terms.
If you don’t know how to multiply variables or how to combine like terms, check out the linked pages.
Example
Multiply:
\[(5x^4-2x^3+6x-9)(3x^2-8x+7)\]
How many columns and rows should I make for my table?
I’ll start by using the terms of the first polynomial to label the columns of my table. Then I’ll use the terms of the second polynomial to label the rows of my table.
It doesn’t really matter which of the two polynomials you use to label the rows and columns, but I like using the longer polynomial for the columns because it saves space on my paper.
Next, I’ll fill each cell in the table by multiplying the row term by the column term.
\((5x^4)(3x^2)=\) \(15x^6\)
\((5x^4)(-8x)=\) \(-40x^5\)
…
\((-9)(-8x)=\) \(72x\)
\((-9)(7)=\) \(-63\)
\(x^6\) terms: \(-15x^6=-15x^6\)
\(x^5\) terms: \(-6x^5-40x^5=-46x^5\)
\(x^4\) terms: \(16x^4+35x^4=51x^4\)
\(x^3\) terms: \(18x^3-14x^3=4x^3\)
\(x^2\) terms: \(-27x^2-48x^2=-75x^2\)
\(x\) terms: \(72x+42x=114x\)
Constants: \(-63=-63\)
\((5x^4-2x^3+6x-9)(3x^2-8x+7)\)
\(=\)
\(-15x^6-46x^5+51x^4+4x^3-75x^2+114x-63\)
Why It Works
When you’re multiplying polynomials, you have to make sure that each term in the first polynomial is multiplied by each term in the second polynomial.
The box method is a great way to do that because it organizes all of the terms into a table. This structured format will help you avoid mistakes like skipping a term or accidentally multiplying two terms from the same polynomial.
It’s also nice because the like terms often show up in diagonal patterns so they’re easy to find when you need to combine them.
