Intro to Polynomials
So, polynomial comes from poly which means many, and nomial meaning term. So, really it just says many terms. Polynomials can have exponents like the “4” in “x to the fourth”, they can have variables like x and y, and they can have constants like 7,000, -3 million, ¼, 1, etc.. You can even combine all of these using addition, subtraction, multiplication and division. HOWEVER a polynomial can never be divided by a variable, only by a constant.
-28 divided by 7x + 1 is not a polynomial, because there is a variable in the denominator.
Now we have to be careful, and remember all of our algebra tricks. Take a look at this.
-28 divided by 7x to the -2 +1 (-28/((7x^-2)+1) ) Because the exponent in the bottom has a negative exponent we know that, when simplified, the exponent is actually moved to the top. Which means that this is a polynomial.
Let’s look at another example:
This is not a polynomial, because when simplified the variable y is moved from the numerator to the denominator,because of the negative exponent; and a polynomial cannot have a variable in the denominator.
Now, try this one:
7xy^24 +28x -7y^23 – 3
Polynomial or not?
This one is a polynomial! It does not have a variable in the denominator.
Another thing that is important to know, is that a polynomial can also just have one term. For instance, 5 alone is a polynomial. This may seem confusing, given that polynomial actually says “many terms,” so I want to make that clear.
Polynomials with 1, 2, or 3 terms all have special names. The name for a polynomial with 1 term is a monomial, a polynomial with 2 terms is a binomial, and a polynomial with 3 terms is a trinomial. There is also the quadrinomial with 4 terms, and the quintinomial with 5 terms; but those are not often used.
A Polynomial can have as many terms you need, but it cannot have an infinite number of terms.
Polynomials are really great in math, because they are easy to work with. If you add polynomials together you are still left with a polynomial. If you multiply polynomials together, you still have a polynomial.
You can divide polynomials; however, the result may not be a polynomial.
Also, Polynomials with one variable, are great for graphing; because their lines are always smooth and continuous.
The last thing you need to know about a polynomial is the standard way to write a polynomial. The standard way to write a polynomial is in descending order, with the highest degree first (farthest to the left), and the lowest degree last. Degrees, in this sense, refer to the number in the exponent.
So, to put the polynomial, 7x – 7 + 24x^3 + 1x^7, you would move the term with the highest number in the exponent to the front. In this case you would move 1x^7 to the front, and so on. Once you have completed, the standard form of this polynomial is 1x^7 + 24x^3 + 7x – 7.
I hope this video helped you to better understand what a polynomial is. Be sure to subscribe to our channel HERE for more videos.
See you next time!