Before adding and subtracting polynomials or multiplying polynomials, it is important to know the definition of a polynomial and polynomial vocabulary. Important polynomial definitions include terms including monomial, the degree of a monomial, polynomial degree and standard form.
When you guys get to the part of your class where you're setting polynomials you're ready for a pretty good chapter. It's a chapter that for many students is not too difficult as long as they can remember some vocabulary. This is what we're going to talk about here, there's a of vocabulary involved with polynomials. First thing is we got to figure out what a monomial is, a monomial is a number, variable or product with non-fractional exponents. So it might look like this, boom that's a monomial. Or I could write this, again it's a monomial, that guy is a product meaning things are being multiplied together. You could also have negative numbers something like that or you could have fractions in your constant terms, x to the 100th doesn't matter. As long as you have products without plus signs or minus signs and as long as you don't have any negatives or fractional exponents you have a monomial.
Let me show you some things that are not monomials if I have something like 4 to the negative 6, negative exponent bad not a monomial. Sometimes negative exponents as you guys might know means you write it in the bottom of the fraction. So if I had something like x to the negative third which is also written as 1 over x to the third, that's not a monomial because we have a negative exponent going on, bad not a monomial. One other thing you might see is something with a square root, like if I had 2 to the one half power, that's a fractional exponent not a monomial. This is what monomials do look like, one other thing that's not a monomial is anything that has a plus or minus sign, x plus y not a monomial.
Okay let's talk about a polynomial, that x plus y business was a polynomial. Here is what it means, a polynomial is the sum or difference of monomials. You don't have to have a sum or difference let me tell you what I'm talking about, the number 3 we already talked about how that's a monomial. A monomial is a type of polynomial, if I have 2 things being added together like 4x+2y that's called a binomial. You guys might already know that mono means 1, bi means 2 think about what it means 3, do you know? Yeah trinomial, what do you think trinomial looks like, any thoughts. Trinomial, trinomial is when you have 3 things being added, 3 things, things meaning monomials being added or subtracted together. So I could have 4xy take away 5x+8 something like that. That's a trinomial because it has 3 terms. These are all special types of polynomials I could also have like 18 things, terms being added or subtracted together that would still be called a polynomial. These are the guys that have special names monomials means 1 term, binomial means 2 terms, trinomial means 3 terms.
Notice we're talking about how many terms not how many variables there are. Like this guys is called the trinomial even though it only involves 2 letters, it only involves x's and y's. It's called a trinomial because there are 3 chunkers of 3 terms being added together. So that's what polynomials look like, a lot of times you have more than 3 things it's just called a polynomial doesn't have a special name. Okay another thing you need to think about is what's called a "degree." The degree of a monomial is the sum of the exponents of the variables. So let's just take my monomial here 4xy the degree would be the sum of the exponents of the variables. Little secret one there, little secret one there so my degree would be the sum of those or 2 that's the degree of that monomial. Or if I had 4x squared y to the third the degree would be 5, you add together those exponents. One other thing is that sometimes there's no exponents at all or no variables so like just say I had the monomial 3, we know 3 is a monomial. What's the degree of that guy? Well since there's no variable term I'm just going to say the degree is equal to zero, degree equals zero and if you wanted to you could you could think of it as 3 times x to the zero because anything to the zero power is 1. That's how we know that the degree of just a constant is equal to zero.
Okay I'm going to back up over here show you some more definitions, next thing we're going to talk about is the degree of a polynomial. We just said the degree of a monomial this is the degree of a polynomial. The degree of a polynomial is the degree of the term with the largest degree. That's tricky cause degrees shows up in that sentence like 3 times. Here's what I mean if we have like 4x to the third plus x squared plus x to the eighth the degree would be the degree of the term with the largest degree. So looking at my terms the one with the largest degree is right here, so the degree of that polynomial is 8.
Pretty much the simplified way of remembering the degree of the polynomial is to remember the degree of the polynomial is the highest exponent number there it is. Standard form of a polynomial is the list of monomials in order from largest to smallest degree or exponent largest or smallest exponent. So this trinomial right here is not in standard form because my largest degree term is this guy here, I wanted to write it in standard form that x to the eighth business needs to come first. My next largest degree is this guy because the exponent is 3 and then my smallest degree for this trinomial is that x squared term. This is what's called standard form because my exponents go in order from highest to lowest.
Last but not least is the term "leading coefficient" forgot the underline it. Leading coefficient is the coefficient of the term with the largest degree. Well if something is in standard form it's really easy to find that because your term with largest degree shows up first. So when this standard form polynomial my leading coefficient would be 1. Sometimes it's a negative number, it's whatever shows up in front of this highest exponent term.
So when you guys get into polynomials you'll see the Math isn't too tricky the hardest part for many students is remembering all this vocabulary so before you start doing your homework problems spend sometime just reviewing this stuff. Get this into your head and I promise it will make your future homework assignments and problems go a lot more quickly.