Polynomial Rules: What Defines Polynomials?
What are the rules for polynomials? The short answer is that polynomials cannot contain the following: division by a variable, negative exponents, fractional exponents, or radicals.
What is a polynomial?
A polynomial is an expression containing two or more algebraic terms. They are often the sum of several terms containing different powers (exponents) of variables.
There are some pretty cool things about polynomials. For example, if you add or subtract polynomials, you get another polynomial. If you multiply them, you get another polynomial.
Polynomials often represent a function. And if you graph a polynomial of a single variable, you'll get a nice, smooth, curvy line with continuity (no holes.)
What does 'polynomial' mean?
The "poly" in polynomial comes from Greek and means "multiple." "Nomial", also Greek, refers to terms, so polynomial means "multiple terms."
The elements of a polynomial
What Makes Up Polynomials
A polynomial is an algebraic expression made up of two or more terms. Polynomials are composed of some or all of the following:
- Variables - these are letters like x, y, and b
- Constants - these are numbers like 3, 5, 11. They are sometimes attached to variables, but can also be found on their own.
- Exponents - exponents are usually attached to variables, but can also be found with a constant. Examples of exponents include the 2 in 5² or the 3 in x³.
- Addition, subtraction, multiplication, and division - For example, you can have 2x (multiplication), 2x+5 (multiplication and addition), and x-7 (subtract.)
Rules: What ISN'T a Polynomial
There are a few rules as to what polynomials cannot contain:
Polynomials cannot contain division by a variable.
For example, 2y2+7x/4 is a polynomial, because 4 is not a variable. However, 2y2+7x/(1+x) is not a polynomial as it contains division by a variable.
Polynomials cannot contain negative exponents.
You cannot have 2y-2+7x-4. Negative exponents are a form of division by a variable (to make the negative exponent positive, you have to divide.) For example, x-3 is the same thing as 1/x3.
Polynomials cannot contain fractional exponents.
Terms containing fractional exponents (such as 3x+2y1/2-1) are not considered polynomials.
Polynomials cannot contain radicals.
For example, 2y2 +√3x + 4 is not a polynomial.
How to find the degree of a polynomial
To find the degree of a polynomial, write down the terms of the polynomial in descending order by the exponent. The term whose exponents add up to the highest number is the leading term. The sum of the exponents is the degree of the equation.
Example: Figure out the degree of 7x2y2+5y2x+4x2.
Start out by adding the exponents in each term.
The exponents in the first term, 7x2y2 are 2 (from 7x2) and 2 (from y2) which add up to four.
The second term (5y2x) has two exponents. They are 2 (from 5y2) and 1 (from x, this is because x is the same as x1.) The exponents in this term add up to three.
The last term (4x2) only has one exponent, 2, so its degree is just two.
Since the first term has the highest degree (the 4th degree), it is the leading term. The degree of this polynomial is four.
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Different types of polynomials
There are different ways polynomials can be categorized. They can be named for the degree of the polynomial as well as by the number of terms it has. Here are some examples:
- Monomials - these are polynomials containing only one term ("mono" means one.) 5x, 4, y, and 5y4 are all examples of monomials.
- Binomials - these are polynomials that contain only two terms ("bi" means two.) 5x+1 and y-7 are examples of binomials.
- Trinomials - a trinomial is a polynomial that contains three terms ("tri" meaning three.) 2y+5x+1 and y-x+7 are examples of trinomials.
There are quadrinomials (four terms) and so on, but these are usually just called polynomials regardless of the number of terms they contain. Polynomials can contain an infinite number of terms, so if you're not sure if it's a trinomial or quadrinomial, you can just call it a polynomial.
A polynomial can also be named for its degree. If a polynomial has the degree of two, it is often called a quadratic. If it has a degree of three, it can be called a cubic. Polynomials with degrees higher than three aren't usually named (or the names are seldom used.)
Operations on Polynomials
Now that you understand what makes up a polynomial, it's a good idea to get used to working with them. If you're taking an algebra course, chances are you'll be doing operations on polynomials such as adding them, subtracting them, and even multiplying and dividing polynomials (if you're not already doing so.)
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