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High School Algebra I: Help and Review25 chapters | 292 lessons

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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.

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 3*x* + 5, our first term is 3*x*, and our second term is 5. Terms are separated by either **addition** or **subtraction**. In our first example, notice how the 3*x* 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 5*x*+3*y*, and our second binomial is 4*x*+7*y*. 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 (5*x* and 4*x*) together because they are like terms. We can do the same for the second terms (3*y* and 7*y*). To combine like terms, we perform the addition or subtraction to the numbers and maintain the variable to its exponent. So, 5*x* + 4*x* = 9*x*, and 3*y* + 7*y* = 10*y*.

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).

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.

I use the **distributive property** when multiplying across the parentheses. I've written my answer starting with the term that has the highest exponent followed by the term with the next highest and so on until the last term with the lowest exponent.

The special case, when your answer will be a binomial, is when you are multiplying two binomials whose first terms and second terms are the same. The only difference between the two binomials is that one has subtraction separating its two terms and the other has addition separating its two terms. When you multiply two such binomials together, you'll see some like terms **canceling each other out** so that your answer is a binomial.

Notice how the first terms of both binomials are the same (*x* and *x*). The second terms are also the same (3 and 3). The only difference between the two binomials is the sign between the terms of each. The first binomial (*x* - 3) has a subtraction sign. The second binomial (*x* + 3) has an addition sign. The like terms that cancel each other out are the 3*x* and the -3*x*. Combine them, and you get 0, which we don't write.

**Binomials** are a special sub-category of polynomials. They only have two terms separated by either addition or subtraction. Adding binomials together requires the combining of like terms. The only way to get a binomial for an answer when adding is if the first and second terms of both binomials are like terms. You multiply binomials by using the distributive property across the parentheses. Most times, you'll not get a binomial for an answer. The only case when you'll get a binomial as an answer is if the two binomials have the same first and second terms and where one binomial has an addition and the other a subtraction.

- A binomial is a mathematical expression containing two terms, which must be separated by addition or subtraction.
- To add binomials, you combine like terms to get an answer.
- To multiply binomials, you use the distributive property.
- Most of the time, you won't get a binomial answer with multiplication.

After viewing this lesson, you should be able to identify binomials and what mathematical operations can be used to solve them.

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High School Algebra I: Help and Review25 chapters | 292 lessons

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