Roots and Powers of Algebraic Expressions

Special Product: Definition & Formula

In this lesson, we will review what a binomial is and how to multiply two binomials together. We will then discuss special cases called special products.

Binomials and FOIL

Binomials are polynomials with two terms. For example:

x + 9 7x - 2y

The two terms in the first binomial are x and 9, and the two terms in the second binomial are 7x and -2y. Binomials are fun to work with in mathematics, because they take on many patterns when we combine them together using various operations. In this lesson, we are going to concentrate on multiplying binomials.

Multiplying binomials may sound like a monumental task, but it's quite easy when we use the FOIL method. FOIL stand for first, inner, outer, last. You are probably wondering what this has to do with multiplying binomials! Well, when we multiply two binomials, like this:

(x + y)(r + s)

We do so by multiplying the first terms of each binomial together to get xr. Then we multiply the outer two terms together to get xs. Next , we multiply the inner two terms together to get yr. Lastly, we multiply the last two terms of each binomial together to get ys, then we add all of these products together to get:

(x + y)(r + s) = xr + xs + yr + ys.

Pretty simple, right? This method is illustrated in the following image.

This process will work for any two binomials, but there are a few special cases where we can simplify this formula even further. These special cases are called special products because they are special cases of products of binomials.

Multiplying the Binomial a + b By Itself

The first special product we will look at is multiplying a binomial by itself. That is:

(a + b)^2 = (a + b)(a + b)

When we perform the FOIL method to this case, we get:

(a + b)(a + b) = a^2 + ab + ab + b^2 = a^2 + 2ab + b^2

This is illustrated in the following image.

We can use this formula anytime we are multiplying a binomial of the form a + b by itself. For example, consider this binomial:

2x + 1

We use the formula to multiply 2x + 1 by itself to get:

(2x + 1)(2x + 1) = (2x)(2x) + 2(2x*1) + 1*1 = 4x^2 + 4x +1

This formula makes things even simpler, don't you agree? Let's look at another special product.

Multiplying the Binomial a - bBy Itself

Another special product is when we multiply a binomial of the form a - b by itself. We can use the FOIL method then simplify to find a nice formula for this special product. We get:

(a - b)(a - b) = a^2 - ab - ab + b^2 = a^2 - 2ab + b^2

We see this illustrated in the image below.

Another formula that makes things easier - yes, please! Consider the example of multiplying the binomial x - 7 by itself. We can use our formula to get:

(x - 7)(x - 7) = x * x - 2(x * 7) + 7^2 = x^2 - 14x + 49

Multiplying a + b By a – b

The last special product is when we are multiplying two binomials together of the form a + b and a - b. Let's use the FOIL method to see what happens when we multiply these two binomials together. That is:

(a + b)(a - b) = a^2 - ab + ab - b^2 = a^2 - b^2

This is great! This formula is even simpler than the others! We see that the two middle terms of the product cancel out leaving us with the formula:
(a + b)(a - b) = a^2 - b^2
We can use this anytime we multiply two binomials of the form (a + b)(a - b). For instance, consider these binomials:
3x + 2
3x – 2
(3x + 2)(3x - 2) = (3x)(3x) - 2*2 = 9x^2 – 4

Lesson Summary

Let's review! Special products are simply special cases of multiplying certain types of binomials together. We have three special products:
• (a + b)(a + b)
• (a - b)(a - b)
• (a + b)(a - b)

These special product formulas are as follows:
• (a + b)(a + b) = a^2 + 2ab + b^2
• (a - b)(a - b) = a^2 - 2ab + b^2
• (a + b)(a - b) = a^2 - b^2

These formulas can be derived using the FOIL method for multiplying binomials, which consists of finding the products of the first two terms of the binomials, the outer terms, the inner terms, and the last two terms, then adding all of these products together. When we want to multiply two binomials together and we are lucky enough to have those binomials in the form of our special products, our job of multiplying those binomials together is much easier.