We now have three properties for multiplying expressions with exponents. Let’s summarize them and then we’ll do some examples that use more than one of the properties.

 

 

 

 

 

 

 

 

 

Notice that in the first monomial, the exponent was outside the parentheses and it applied to both factors inside. In the second monomial, the exponent was inside the parentheses and so it only applied to the n.

 

 

 

 

Multiply Monomials

Since a monomial is an algebraic expression, we can use the properties for simplifying expressions with exponents to multiply the monomials.

 

 

 

 

 

For more examples of how to use the power and product rules of exponents to simplify and multiply monomials, watch the following video.

Multiply a Monomial by a Polynomial

In Distributive Property you learned to use the Distributive Property to simplify expressions such as [latex]2\left(x - 3\right)[/latex]. You multiplied both terms in the parentheses, [latex]x\text{ and }3[/latex], by [latex]2[/latex], to get [latex]2x - 6[/latex]. With this chapter’s new vocabulary, you can say you were multiplying a binomial, [latex]x - 3[/latex], by a monomial, [latex]2[/latex]. Multiplying a binomial by a monomial is nothing new for you!

 

 

 

 

 

 

 

 

 

Multiplying a monomial by a trinomial works in much the same way.

 

 

 

 

 

 

Now we will have the monomial as the second factor.

 

 

In the following video we show more examples of how to multiply monomials with other polynomials.

Multiply Two Binomials

Just like there are different ways to represent multiplication of numbers, there are several methods that can be used to multiply a binomial times a binomial.

Using the Distributive Property

We will start by using the Distributive Property. Look again at the following example.

We distributed the [latex]p[/latex] to get	[latex]x\color{red}{p}+3\color{red}{p}[/latex]
What if we have [latex]\left(x+7\right)[/latex] instead of [latex]p[/latex] ?
Distribute [latex]\left(x+7\right)[/latex] .
Distribute again.	[latex]{x}^{2}+7x+3x+21[/latex]
Combine like terms.	[latex]{x}^{2}+10x+21[/latex]

Notice that before combining like terms, we had four terms. We multiplied the two terms of the first binomial by the two terms of the second binomial—four multiplications.
Be careful to distinguish between a sum and a product.
[latex]\begin{array}{cccc}\hfill \mathbf{\text{Sum}}\hfill & & & \hfill \mathbf{\text{Product}}\hfill \\ \hfill x+x\hfill & & & \hfill x\cdot x\hfill \\ \hfill 2x\hfill & & & \hfill {x}^{2}\hfill \\ \hfill \text{combine like terms}\hfill & & & \hfill \text{add exponents of like bases}\hfill \end{array}[/latex]

 

 

 

Now we’ll see how to multiply binomials where the variable has a coefficient.

 

 

In the previous examples, the binomials were sums. When there are differences, we pay special attention to make sure the signs of the product are correct.

 

 

 

Up to this point, the product of two binomials has been a trinomial. This is not always the case.

 

 

To see another example of how to visualize multiplying two binomials, watch the following video. We use an area model as well as repeated distribution to multiply 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.

In the video that follows, you will see another example of using a table to multiply two binomials.

Multiply a Trinomial by a Binomial

We have multiplied monomials by monomials, monomials by polynomials, and binomials by binomials. Now we’re ready to multiply a trinomial by a binomial. Remember, the FOIL method will not work in this case, but we can use either the Distributive Property or the Vertical Method. We first look at an example using the Distributive Property.

 

 

 

Now let’s do this same multiplication using the Vertical Method.

 

 

 

Watch the following video to see more examples of multiplying polynomials.