7.5+Defining+Polynomials

Section 7.5 - Defining Polynomials
A term is a product of constants and variables (or a single variable or constant). Examples include: 5, w, 3xy.

Polynomials are, (and I over-generalize here), when you add or subtract one or more terms.There is another rule that says the terms can't have a variable with a negative exponent, essentially a variable as a denominator.

Examples of polynomials:



Polynomial is the vocabulary term that describes all of these, but there are a couple of ways to describe them more specifically.


 * Prefixes**

One way to describe polynomials is by the number of terms.

One term - 7xy - monomial Two terms - 8x + 3 - binomial Three terms - 8xy - 2x + 5 - trinomial Four or more terms - 5xy + 3x - 2y - 99 -- polynomial

Polynomial is also used generically for any number of terms, so a binomial is also a polynomial (but not all polynomials are binomials).


 * Degree of a Polynomial**

Degree refers to the exponents on a variable. Sometimes we say x to the 7th degree to say x with an exponent of 7.

The** degree of a term **is the sum of the exponents on the variables in that term. However, don't forget that a variable without an exponent still really has an unwritten exponent of 1.



The ** degree of a polynomial **is the same as the largest degree of it's terms. If you have a polynomial, find the term with the highest degree and that is also the degree of the polynomial.

Polynomials are usually (should be) written in descending order, meaning the term with the highest degree is usually written first.




 * Number of Terms**

Another way polynomials are described is by counting the number of unique variables in the polynomial.



The top example would be described as "a trinomial in two variables with a degree of 4". The bottom one, "a polynomial in two variables with a degree of 13".


 * Chapter Seven Practice Problems**

**Where to from here?**
7.1 Negative Exponents 7.2 Product Rule 7.3 Power Rule 7.4 Quotient Rule 7.5 Defining Polynomials 7.6 Adding Polynomials 7.7 Subtracting Polynomials 7.8 Multiplying Polynomials 7.9 Dividing Polynomials Chapter Seven Summary Chapter Seven Homework