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Lesson 115: Working with Monomials
In this important lesson, we'll learn what monomials are, and how to work with them. We'll also learn some other new definitions.
First review Lesson 103. Here are some monomials: 17, x, 3a, 8m2, 4xy, 9x2y, 100x2y3. A monomial is either a constant alone, or a variable alone, or a constant multiplied by one or more variables. The variables may be raised to various powers. If one or more variables are multiplied by a constant, such as 4xy, then the constant is called the coefficient of the monomial.
You might be asked to identify the degree of a monomial. The degree is the sum of the exponents of the variable terms. A monomial that is just a constant has a degree of 0, since there are no variables at all. A monomial that is just a plain variable, such as x, has a degree of 1, because it really has an exponent of 1 that we omit. 8m2, has a degree of 2. 100x2y3 has a degree of 5 (we add the exponents).
We can add and subtract like monomials. That means monomials that have the exact same variables and exponents. For example, we can add 3a + 2a to get 5a. We can do 9x2 - 3x2 to get 6x2. We can add 7x2y3 + 8x2y3 to get 15x2y3. We are combining like terms. Think of the like terms as apples, no matter how complicated they are. We are just adding or subtracting apples, to get another quantity of apples.
To multiply a monomial by a constant, just multiply the constant times the coefficient of the monomial, if there is one. For example. 3(7x2) = 21x2. 17(x3y) = 17x3y.
To multiply a monomial by a monomial, multiply the coefficients, then multiply each variable. Remember from previous lessons that when we multiply variables, we actually add the exponents. For example: (5x2y3)(6xy4) = 30x3y7. We multiplied the coefficients, then we added the exponents of the x terms. Remember that x by itself is really x1, so we added 2 + 1. Then we added 3 + 4 for the y terms.
To divide a monomial by a constant, just divide the coefficient by the constant. For example, 12x5 / 4 = 3x5.
To divide a monomial by a monomial, divide the coefficients, and then divide the variables, remembering from previous lessons that when you divide variables, you actually subtract the exponents. For example, 20x5y3 / 5x2y2 = 4x3y. Remember that y1 is typically represented as just y.
To raise a monomial to a power, we raise the coefficient to the power, and then raise each term to the power, remembering our rule for how to do this. For example, (3x2y)4 = 81x8y4. We have 34, and then we had to multiply our variable powers times the power that we were raising the monomial to.
This lesson was a bit tricky. Make sure that you study it carefully, and memorize all of the rules and terms that were presented. More importantly, make sure that you understand how and why the rules work the way that they do.
Remember that you can ask a math question if you have additional questions about a topic, or you can contact me if you have any comments or suggestions for this site.