If you see any of these, then you can zap them out of your balloon as they are not polynomials. Even though polynomials literally means many terms, you can have just one term or just two terms. A term means the multiplication of a number with a variable. The variable's exponent can be any positive whole number. All terms are separated by either a plus or a minus symbol. Of course, you can have as many terms as you want in a polynomial. There is no limit. All of these are polynomials that belong inside the balloon.
Notice how none of the variables have negative exponents, are in the denominator, or are inside a radical. All of the exponents for the variables are positive whole numbers. Also notice how all the terms are separated by either a plus or a minus. All of these polynomials are written in what is called standard form, starting with the largest exponent and working my way down. Now let's talk about binomials.
Binomials
So, inside the balloon are two smaller balloons. One of the smaller balloons is for binomials. Binomialmeans two terms. So, what kinds of polynomials do you think binomials cover? That's right; binomials include all polynomials that only have two terms. So, from what we see inside the big polynomial balloon, the two polynomials that are binomials are the x + 7 and the x^4 + 3x. Both of these only have two terms and that is all that is needed for them to go inside the smaller balloon. Now what about quadratics?
Quadratics
We have another small balloon inside the large balloon and it is for quadratics. Quadratics include polynomials whose highest exponent is 2. We can have quadratics with only one term, two terms, or three terms. But three terms is the maximum limit. We can't have more than that. The one polynomial that we see that fits this criteria is the x^2 + 3x - 1. Other quadratics that we can add to our little balloon includes these.
Do you see how all of these are polynomials whose highest exponent is 2 and that all of these are either one term, two, or three terms in length? They all fit the criteria for quadratics. It is definitely possible that you can have a polynomial that can belong to two or all three balloons.
Lesson Summary
To summarize everything we have learned, polynomials means many terms, binomials means two terms, and quadratics means polynomials whose highest exponent is 2. All of these are polynomials with binomials and quadratics being special cases. The three restrictions that make polynomials easy to work with are that:
- The exponents cannot be negative.
- The variables cannot be in the denominator.
- The variables cannot be inside a radical.