10: Polynomials
- Page ID
- 114997
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Expressions known as polynomials are used widely in algebra. Applications of these expressions are essential to many careers, including economists, engineers, and scientists. In this chapter, we will find out what polynomials are and how to manipulate them through basic mathematical operations.
- 10.2: Add and Subtract Polynomials
- In this section, we will work with polynomials that have only one variable in each term. The degree of a polynomial and the degree of its terms are determined by the exponents of the variable. Working with polynomials is easier when you list the terms in descending order of degrees. When a polynomial is written this way, it is said to be in standard form. Adding and subtracting polynomials can be thought of as just adding and subtracting like terms.
- 10.3: Use Multiplication Properties of Exponents
- In this section, we will begin working with variable expressions containing exponents. Remember that an exponent indicates repeated multiplication of the same quantity. You have seen that when you combine like terms by adding and subtracting, you need to have the same base with the same exponent. But when you multiply and divide, the exponents may be different, and sometimes the bases may be different, too. We’ll derive the properties of exponents by looking for patterns in several examples.
- 10.4: Multiply Polynomials
- In this section, we will begin multiplying polynomials with degree one, two, and/or three. Just like there are different ways to represent multiplication of numbers, there are several methods that can be used to multiply a polynomial by another polynomial. The Distributive Property is the first method that you have already encountered and used to find the product of any two polynomials.
- 10.5: Divide Monomials
- In this section, we will look at the exponent properties for division. A special case of the Quotient Property is when the exponents of the numerator and denominator are equal. It leads us to the definition of the zero exponent, which states that if a is a non-zero number, then a^0 = 1. Any nonzero number raised to the zero power is 1. The quotient to a power property of exponents states that to raise a fraction to a power, you raise the numerator and denominator to that power.
- 10.6: Integer Exponents and Scientific Notation
- The negative exponent tells us to re-write the expression by taking the reciprocal of the base and then changing the sign of the exponent. Any expression that has negative exponents is not considered to be in simplest form. We will use the definition of a negative exponent and other properties of exponents to write an expression with only positive exponents.
- 10.7: Introduction to Factoring Polynomials
- Earlier we multiplied factors together to get a product. Now, we will be reversing this process; we will start with a product and then break it down into its factors. Splitting a product into factors is called factoring. In The Language of Algebra we factored numbers to find the least common multiple (LCM) of two or more numbers. Now we will factor expressions and find the greatest common factor of two or more expressions. The method we use is similar to what we used to find the LCM.
Figure 10.1 - The paths of rockets are calculated using polynomials. (credit: NASA, Public Domain)
Contributors and Attributions
Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/fd53eae1-fa2...49835c3c@5.191."