Learning Objectives
- Apply the FOIL method to multiply two binomials
- Use a table to multiply two binomials
- Simplify the product of two binomials given a wide variety of variables, constants, signs, and arrangement of terms in the binomial
Foil Crane
In the last section we finished with an example of multiplying two binomials,[latex]\left(x+4\right)\left(x+2\right)[/latex]. In this section we will provide examples of how to use two different methods to multiply to binomials. Keep in mind as you read through the page that simplify and multiply are used interchangeably.
FOIL
Some people use the FOIL method to keep track of which pairs of terms have been multiplied when you are multiplying two binomials. This is not the same thing you use to wrap up leftovers, but an acronym for First, Outer, Inner, Last. Let’s go back to the example from the previous page, where we were asked to multiply the two binomials: [latex]\left(x+4\right)\left(x+2\right)[/latex]. The following steps show you how to apply this method to multiplying two binomials.
[latex]\begin{array}{l}\text{First}\text{ term in each binomial}: \,\,\,\,\,\,\,\,\,\,\,\,\,\,\left(x+4\right)\left(x+2\right)\,\,\,\,\,\,\,\,\,\,\,\,\,x\left(x\right)=x^{2}\\\text{Outer terms}:\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left(x+4\right)\left(x+2\right)\,\,\,\,\,\,\,\,\,\,\,\,\,x\left(2\right)=2x\\\text{Inner terms}:\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left(x+4\right)\left(x+2\right)\,\,\,\,\,\,\,\,\,\,\,\,\,4\left(x\right)=4x\\\text{Last terms in each binomial}:\,\,\,\,\,\,\,\,\,\,\,\,\,\left(x+4\right)\left(x+2\right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,4\left(2\right)=8\end{array}[/latex]
When you add the four results, you get the same answer, [latex]x^{2}+2x+4x+8=x^{2}+6x+8[/latex].
The last step in multiplying polynomials is to combine like terms. Remember that a polynomial is simplified only when there are no like terms remaining.
Caution! Note that the FOIL method only works for multiplying two binomials together. It sill not work for multiplying a binomial and a trinomial, or two trinomials.
Order Doesn’t Matter When You Multiply
One of the neat things about multiplication is that terms can be multiplied in either order. The expression [latex]\left(x+2\right)\left(x+4\right)[/latex] has the same product as [latex]\left(x+4\right)\left(x+2\right)[/latex], [latex]x^{2}+6x+8[/latex]. (Work it out and see.) The order in which you multiply binomials does not matter. What matters is that you multiply each term in one binomial by each term in the other binomial.
Polynomials can take many forms. So far we have seen examples of binomials with variable terms on the left and constant terms on the right, such as this binomial [latex]\left(2r-3\right)[/latex]. Variables may also be on the right of the constant term, as in this binomial [latex]\left(5+r\right)[/latex]. In the next example, we will show that multiplying binomials in this form requires one extra step at the end.
Example
Find the product.[latex]\left(3–s\right)\left(1-s\right)[/latex]
In the next example, you will see that sometimes there are constants in front of the variable. They will get multiplied together just as we have done before.
Example
Simplify [latex]\left(4x–10\right)\left(2x+3\right)[/latex] using the FOIL acronym.
The video that follows gives another example of multiplying two binomials using the FOIL acronym. Remember this method only works when you are multiplying two binomials.
The Table Method
You may see a binomial multiplied by itself written as [latex]{\left(x+3\right)}^{2}[/latex] instead of [latex]\left(x+3\right)\left(x+3\right)[/latex]. To find this product, let’s use another method. We will place the terms of each binomial along the top row and first column of a table, like this:
| [latex]x[/latex] | [latex]+3[/latex] | |
| [latex]x[/latex] | ||
| [latex]+3[/latex] |
Now multiply the term in each column by the term in each row to get the terms of the resulting polynomial. Note how we keep the signs on the terms, even when they are positive, this will help us write the new polynomial.
| [latex]x[/latex] | [latex]+3[/latex] | |
| [latex]x[/latex] | [latex]x\cdot{x}=x^2[/latex] | [latex]3\cdot{x}=+3x[/latex] |
| [latex]+3[/latex] | [latex]x\cdot{3}=+3x[/latex] | [latex]3\cdot{3}=+9[/latex] |
Now we can write the terms of the polynomial from the entries in the table:
[latex]\left(x+3\right)^{2}[/latex]
= [latex]x^2[/latex] + [latex]3x[/latex] + [latex]3x[/latex] + [latex]9[/latex]
= [latex]x^{2}[/latex] + [latex]6x[/latex] + [latex]9[/latex].
Pretty cool, huh?
So far, we have shown two methods for multiplying two binomials together. Why are we focusing so much on binomials? They are one of the most well studied and widely used polynomials, so there is a lot of information out there about them. In the previous example, we saw the result of squaring a binomial that was a sum of two terms. In the next example we will find the product of squaring a binomial that is the difference of two terms.
Example
Square the binomial difference [latex]\left(x–7\right)[/latex]
Caution! It is VERY important to remember the caution from the exponents section about squaring a binomial:
You can’t move the exponent into a grouped sum because of the order of operations!!!!!
INCORRECT: [latex]\left(2+x\right)^{2}\neq2^{2}+x^{2}[/latex]
CORRECT: [latex]\left(2+x\right)^{2}=\left(2+x\right)\left(2+x\right)[/latex]
In the video that follows, you will see another examples of using a table to multiply two binomials.
Further Examples
The next couple of examples show you some different forms binomials can take. In the first, we will square a binomial that has a coefficient in front of the variable, like the product in the first example on this page. In the second we will find the product of two binomials that have the variable on the right instead of the left. We will use both the FOIL method and the table method to simplify.
Example
Find the product. [latex]\left(2x+6\right)^{2}[/latex]
In the last example, we want to show you another common form a binomial can take, each of the terms in the two binomials is the same, but the signs are different. You will see that in this case, the middle term will disappear.
Example
Multiply the binomials. [latex]\left(x+8\right)\left(x–8\right)[/latex]
Think About It
There are predictable outcomes when you square a binomial sum or difference. In general terms, for a binomial difference,
[latex]\left(a-b\right)^{2}=\left(a-b\right)\left(a-b\right)[/latex],
the resulting product, after being simplified, will look like this:
[latex]a^2-2ab+b^2[/latex].
The product of a binomial sum will have the following predictable outcome:
[latex]\left(a+b\right)^{2}=\left(a+b\right)\left(a+b\right)=a^2+2ab+b^2[/latex].
Note that a and b in these generalizations could be integers, fractions, or variables with any kind of constant. You will learn more about predictable patterns from products of binomials in later math classes.
In this section we showed two ways to find the product of two binomials, the FOIL method, and by using a table. Some of the forms a product of two binomials can take are listed here:
- [latex]\left(x+5\right)\left(2x-3\right)[/latex]
- [latex]\left(x+7\right)^{2}[/latex]
- [latex]\left(x-1\right)^{2}[/latex]
- [latex]\left(2-y\right)\left(5+y\right)[/latex]
- [latex]\left(x+9\right)\left(x-9\right)[/latex]
- [latex]\left(2x-4\right)\left(x+3\right)[/latex]
And this is just a small list, the possible combinations are endless. For each of the products in the list, using one of the two methods presented here will work to simplify.
Summary
Multiplication of binomials and polynomials requires an understanding of the distributive property, rules for exponents, and a keen eye for collecting like terms. Whether the polynomials are monomials, binomials, or trinomials, carefully multiply each term in one polynomial by each term in the other polynomial. Be careful to watch the addition and subtraction signs and negative coefficients. A product is written in simplified form if all of its like terms have been combined.