Question Video: Expanding the Product of Two Binomials Mathematics • 9th Grade

Video Transcript

Expand the product two 𝑚 plus 𝑛 multiplied by two 𝑚 minus 𝑛.

The expression we’ve been asked to expand is the product of two binomials: two 𝑚 plus 𝑛 and two 𝑚 minus 𝑛. To expand this product means we need to multiply out the brackets, also known as distributing the parentheses. When we do so, we need to ensure we multiply each term in the first binomial by each term in the second. We can do this in a number of different ways. But the method we’ll demonstrate here is called the vertical method. This is similar to the column method for multiplying integers.

We begin by writing one factor below the other and then find the product of each pair of terms. We start by multiplying each term in the binomial two 𝑚 plus 𝑛 by negative 𝑛. Two 𝑚 multiplied by negative 𝑛 is negative two 𝑚𝑛, and 𝑛 multiplied by negative 𝑛 is negative 𝑛 squared. Next, we multiply each term in the binomial two 𝑚 plus 𝑛 by the other term in the second binomial, two 𝑚. Two 𝑚 multiplied by two 𝑚 is four 𝑚 squared, and 𝑛 multiplied by two 𝑚 is two 𝑚𝑛.

We now add these four terms together. Note that negative two 𝑚𝑛 plus two 𝑚𝑛 is zero. So these terms cancel, and the sum is four 𝑚 squared minus 𝑛 squared. Therefore, we’ve found that the expanded form of the product of two 𝑚 plus 𝑛 and two 𝑚 minus 𝑛 is four 𝑚 squared minus 𝑛 squared.

It’s worth noting that this expression can also be written as two 𝑚 all squared minus 𝑛 squared. This is known as a difference of two squares. This question illustrates the general result that the product of 𝑥 plus 𝑦 and 𝑥 minus 𝑦 is always equal to 𝑥 squared minus 𝑦 squared.