Find the product of monomials

Multiplying polynomials involves applying the rules of exponents and the distributive property to simplify the product. Polynomial multiplication can be useful in modeling real world situations. Understanding polynomial products is an important step in learning to solve algebraic equations involving polynomials. There are many, varied uses for polynomials including the generation of 3D graphics for entertainment and industry, as in the image below.

Surfaces made from polynomials with AutoCAD

In the exponents section, we practiced multiplying monomials together, like we did with this expression: [latex]24{x}^{8}\cdot2{x}^{5}[/latex]. The only thing different between that section and this one is that we called it simplifying, and now we are calling it polynomial multiplication.  Remember that simplifying a mathematical expression means performing as many operations as we can until there are no more to perform, including multiplication.  In this section we will show examples of how to multiply more than just monomials.  We will multiply monomials with binomials and trinomials. We will also learn some techniques for multiplying two binomials together.

That’s it! When multiplying monomials, multiply the coefficients together, and then multiply the variables together. Remember, if two variables have the same base, follow the rules of exponents, like this:

[latex] \displaystyle 5{{a}^{4}}\cdot 7{{a}^{6}}=35{{a}^{10}}[/latex]

The following video provides more examples of multiplying monomials with different exponents.

Find the product of a monomial and a polynomial

The distributive property can be used to multiply a monomial and a binomial. Just remember that the monomial must be multiplied by each term in the binomial. In the next example, you will see how to multiply a second degree monomial with a binomial.  Note the use of exponent rules.

Now let’s add another layer by multiplying a monomial by a trinomial. Consider the expression [latex]2x\left(2x^{2}+5x+10\right)[/latex].

This expression can be modeled with a sketch like the one below.

The only difference between this example and the previous one is there is one more term to distribute the monomial to.

[latex]\begin{array}{c}2x\left(2x^{2}+5x+10\right)=2x\left(2x^{2}\right)+2x\left(5x\right)=2x\left(10\right)\\=4x^{3}+10x^{2}+20x\end{array}[/latex]

You will always need to pay attention to negative signs when you are multiplying. Watch what happens to the sign on the terms in the trinomial when it is multiplied by a negative monomial in the next example.

The following video provides more examples of multiplying a monomial and a polynomial.

Find the product of two binomials

Now let’s explore multiplying two binomials. For those of you that use pictures to learn, you can draw an area model to help make sense of the process. You’ll use each binomial as one of the dimensions of a rectangle, and their product as the area.

The model below shows [latex]\left(x+4\right)\left(x+2\right)[/latex]:

Each binomial is expanded into variable terms and constants, [latex]x+4[/latex], along the top of the model and [latex]x+2[/latex] along the left side. The product of each pair of terms is a colored rectangle. The total area is the sum of all of these small rectangles, [latex]x^{2}+2x+4x+8[/latex], If you combine all the like terms, you can write the product, or area, as [latex]x^{2}+6x+8[/latex].

Find the product of two binomials using the Distributive Property

You can use the distributive property to determine the product of two binomials.

Look back at the model above to see where each piece of [latex]x^{2}+2x+4x+8[/latex] comes from. Can you see where you multiply [latex]x[/latex] by [latex]x + 2[/latex], and where you get [latex]x^{2}[/latex] from [latex]x\left(x\right)[/latex]?

Another way to look at multiplying binomials is to see that each term in one binomial is multiplied by each term in the other binomial. Look at the example above: the [latex]x[/latex] in [latex]x+4[/latex] gets multiplied by both the [latex]x[/latex] and the 2 from [latex]x+2[/latex], and the 4 gets multiplied by both the [latex]x[/latex] and the 2.

The following video provides an example of multiplying two binomials using an area model as well as repeated distribution.

Next we will explore other methods for multiplying two binomials, and become aware of the different forms that binomials can have.

Find the product of two binomials using the FOIL Method

Foil Crane

We just looked at the 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.

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 a previous example, 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 does 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]. They are both equal to [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.

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.

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.

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.  We will also demonstrate how to use another method to multiply binomials.  It is called the Table Method.

Find the product of two binomials using the Table Method

In the next two examples, 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.

Square a binomial

In the next few examples, we will look at what happens when a binomial is squared.

The expression [latex]{\left(x+3\right)}^{2}[/latex] means the same thing as [latex]\left(x+3\right)\left(x+3\right)[/latex].  To find this product, let’s use the table method. We will place the terms of each binomial along the top row and first column of a table, like this:

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.

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].

The following video works through two more examples of squaring a binomial.

We have looked at two methods for multiplying two binomials together, the FOIL method and the Table method.  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. 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.