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### Course: Algebra (all content)>Unit 10

Lesson 3: Adding & subtracting polynomials: two variables

# Polynomials review

Quickly review what polynomials are, common related terms (e.g. degree, coefficient, binomial, etc.), addition & subtraction of polynomials, and modeling area with polynomials. Created by Sal Khan and CK-12 Foundation.

## Want to join the conversation?

• What's a binomial?
• "bi" translates as "two". A binomial is a polynomial with exactly two terms. Some examples are x^2+x, x+3, or y-x, y^6x^4 - 5. A monomial is a polynomial with exactly one term. A polynomial is the sum of any number of terms including just one.

x+3x is not a binomial because you can simplify it to 4x which is a monomial. You should combine all like terms before counting to see how many there are.
• Why do polynomials have to have a positive integer ?
Is this a convention ?
• Do you mean a positive integer as the exponent for its variables? It is part of the definition of polynomial. Once you define a polynomial with those restrictions, then you can apply all sorts of mathematical methods and analyses to a given polynomial.

If instead of a positive integer as the exponents for each term containing a variable, you have fractional exponents or negative exponents, you do not have a polynomial.

You can have an expression with all sorts of terms, but it would not be a polynomial. In calculus it is quite typical to have expressions such as x^(3/2) - 5x^(-1/2). These just are not polynomials. Math works just fine with these non-polynomial expressions as long as you don't try to apply methods that apply only to polynomials..
• We didn't get much info - other than "becuase" - on why fractional and/or negative exponents are excluded from the definition of polynomials.

Can anyone elaborate further?
• The nature of definitions is that they determine what is included and what is not, you don't really need a reason.

But, functions with fractional and/or negative exponents exhibit an entirely different type of behavior than those without, so I don't know why you would WANT to group them together.
• Sal said the coefficient is a constant term, couldn't it be another variable?

Like:
x² + yx + 7

Can't y be called coefficient?
• Yes y can be a coefficient because it is a constant multiple for x
(x is always going to be y times bigger than the original regardless of the number)
• At about , Sal says that 5 is the highest exponent on a variable. Does this mean that a degree can only come from the highest exponent on a variable, or can it be on a normal number as well?
• The degree of a polynomial is by definition the largest exponent of the variable. So, yes we only consider the exponent of the the variable.
So, x² + 50³¹⁷ would still be a second degree polynomial.
• How 'bout a DEKANOMIAL?
OK, that was a bit of a joke, but can you actually use a dekanomial?

As I'm writing this, "dekanomial" currently has the infamous red squiggly line underneath it, so it probably doesn't exist. But does it?
• A Dekanomial is a polynomial with ten terms. It is actually called a Decanomial with an "s" sound. Google doesn't recognize it...
• At about you talked about descending order, what would the degree of a term without an exponent be?
• If the term is something like 2x, then there is an exponent on that variable. If one is not written, it's implied that it is to the 1st power.

If there is no variable at all, like in 4, you would say that it is degree 0.

The only other weird case is that a term of 0 is said to have no degree.
• What is a constant number?
• A constant is, loosely speaking, a quantity that does not change.