# Binomial: Definition & Examples

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• 0:00 What Is a Binomial?
• 1:12 Can You Add Binomials?
• 3:00 Multiplying Binomials
• 3:27 Special Multiplication Case
• 4:23 Lesson Summary

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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Binomial is a little term for a unique mathematical expression. Learn what makes binomials so special, how to easily identify them, and the mathematical operations that can be performed on them. After the lesson, test yourself with a quiz.

## What Is a Binomial?

A binomial is a mathematical expression with two terms.

All of these examples are binomials. Study them for a bit, and see if you can spot a pattern. The following is a list of what binomials must have:

• They must have two terms.
• If the variables are the same, then the exponents must be different.
• Exponents must be whole positive integers. They cannot be negatives or fractions.

A term is a combination of numbers and variables. In the example 3x + 5, our first term is 3x, and our second term is 5. Terms are separated by either addition or subtraction. In our first example, notice how the 3x and 5 are separated by addition. In the last example, we have a binomial whose two terms both have the same variable s. Notice how each term has its variable to a different exponent. The first term has an exponent of 5, and the second term has an exponent of 4. While we can have fractions for our numbers, we cannot have fractional exponents.

Here are some examples of expressions that are not binomials.

Looking at these, we can say that there are things that binomials cannot have.

• Exponents cannot be negatives or fractions.
• And variables cannot be in the denominator.

Now that we can identify binomials, let's see about adding two binomials together. Adding them is fairly straightforward as long as you remember to combine like terms. The caveat here is that many times when you add binomials, your answer won't be a binomial. The only time you will get a binomial back as an answer is if both of your binomials share like terms like this:

Our first binomial is 5x+3y, and our second binomial is 4x+7y. The first term of both binomials have the same variable to the same exponent, x. The second term of both binomials also shares a variable to the same exponent, y. We can go ahead and combine the first terms (5x and 4x) together because they are like terms. We can do the same for the second terms (3y and 7y). To combine like terms, we perform the addition or subtraction to the numbers and maintain the variable to its exponent. So, 5x + 4x = 9x, and 3y + 7y = 10y.

Most times, though, you'll most likely be adding binomials that don't share like terms.

In this example, we end up with an expression that is not a binomial. Why? Because the first term in our first binomial has an exponent of 2, but the first term of our second binomial has an exponent of 1. These cannot be combined because they are not like terms. Hence, we have to keep them separated. The second terms of both binomials are like terms, and we can combine those (3 + 7 = 10).

## Multiplying Binomials

You can multiply two binomials together also. To multiply two binomials together, we take each term of one binomial and multiply it with each term of the other binomial, and then we add everything together and combine as many like terms as we can.

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