Objective Learn what polynomials by knowing how to use the laws of exponents to multiply monomials.
To this point, you surely have worked with linear expressions and equations. Now you need to understand how to do algebra with higher degree expressions, involving higher powers of the variables. The key ideas in this lesson are the laws of exponents, which are used to simplify monomial expressions.
Let's begin by reviewing exponents. If x is a variable, then x 2 denotes x Â· x , x 3 denotes x Â· x Â· x , and more generally, x n denotes
Definition of Monomials
A monomial is a number, a variable, or a product of numbers and variables.
Numbers are referred to as constants.
When there are several occurrences of the same variable, the expression is typically written in exponent form.
Monomials: 5x 2 , 7 xy , -3x 3 , 4y , z 100
Laws of Exponents and Multiplying Monomials
When multiplying monomials, analyze the product of two powers of the same variable, for instance, x 2 Â· x 3 . A good way to illustrate this is by multiplying various powers of 2 together.
Simplify 2 2 Â· 2 3.
2 2 Â· 2 3 = 4 Â· 8 = 32 = 2 5 2
Simplify 2 4 Â· 2 5 .
2 4 Â· 2 5 = 16 Â· 32 = 512 = 2 9
In Examples 1 and 2, the exponent of the result of the multiplication is the sum of the exponents in the two factors. In Example 1, 5 = 2 + 3, and in Example 2, 9 = 4 + 5. Why is this true? Recall that exponents are just shorthand for a repeated product of the same number or variable. So,
In general, when multiplying x m and x n , the result is as follows.
Product of Powers
When a power of x is multiplied by another power of x, the result is a power of x whose exponent is the sum of the exponents of the factors.
x m Â· x n = x m + n
This is true for any number or variable x , and for any whole numbers m and n . Now use this idea to multiply monomials.
Simplify 3x 5 Â· 4x 2 .
Notice that the constants are grouped together and the variables are grouped together.
Simplif 9y 8 Â· ( - y 7 ).
9y 8 Â· ( - y 7 ) = [9 Â· ( -1)] Â· ( y 8 Â· y 7 ) = -9y 15