Multiplication of binomials geometrically

In this lesson, we will perform the multiplication of binomials geometrically. In other words, we will use area models to do the multiplication.

Example #1

Consider the product (x + 4)2

(x + 4)2 = (x + 4)(x + 4). The area model is shown in the figure below.

Area model of (x + 4)(x + 4)


The area we are looking for is that of the preceding entire square, which is the sum of the four inner regions.

(x + 4)2 =  x2 + 4x + 4x + 16

(x + 4)= x2 + 8x + 16

Example #2

Consider the product (x + 3)(x + 2). This is very similar to example #1.

Area model of (x + 3)(x + 2)


The area we are looking for is that of the preceding entire rectangle, which is the sum of the four inner regions.

(x + 3)(x + 2) =  x2 + 3x + 2x + 6

(x + 3)(x + 2) = x2 + 5x + 6

Two more interesting examples showing how to do multiplication of binomials geometrically 

Example #3

Consider the product (x + 4)(x - 2)

Area model of (x + 4)(x - 2)

Notice that the area we are looking for this time though cannot be found by adding the sum of the four inner regions. Only the top two rectangles will give us the area that we are looking since they are the ones that model (x + 4)(x - 2)

(x + 4)(x - 2) =  x(x - 2) + 4(x - 2)

(x + 4)(x - 2) = x2 -2x + 4x - 8

(x + 4)(x - 2) = x2 + 2x - 8

Example #4

Consider the product (x - 4)(x - 2)

The area we are looking for cannot be found by using the upper left rectangle which is a representation of (x-4))(x-2). To find the area we need to find the area of the large square and then subtract the areas of the other three rectangles from it.

(x - 4)(x - 2) = x2 - 4(x - 2) - 2(x - 4) - 8

(x - 4)(x - 2) = x2 - 4x + 8 - 2x + 8 - 8

(x - 4)(x - 2) = x2 - 4x + 8 - 2x

(x - 4)(x - 2) = x2 - 6x + 8 

Recent Articles

  1. Introduction to Physics

    Nov 18, 20 01:20 PM

    Top-notch introduction to physics. One stop resource to a deep understanding of important concepts in physics

    Read More

New math lessons

Your email is safe with us. We will only use it to inform you about new math lessons.

                                 Follow me on Pinterest

Real Life Math Skills

Learn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball.