 Welcome back to our lecture series math 1050 college algebra students at Southern Utah University as usual. I'm your professor today Dr. Andrew Missildine and Our lecture series section 4.2 We reach finally the main topic for this chapter the idea of polynomial functions a polynomial function is a function of the form f of x equals a sub n x of n plus a sub n minus 1 x the n minus 1 plus a sub you know the next terms all the way down to a 1 x plus a sub 0 here And so really what we want to think of as a polynomial is just gonna be a collection of monomial functions So we talked about power functions in the previous video with a special emphasis on monomials Monomial functions being those of the form y equals a times x to the n for some power of x right there And so the word monomial itself really does mean one Nomial the no meal just here. This means name Mono just mean there's one term as opposed to like a polynomial for which there are many terms in the collection there So we combine together all these different monomials and we form polynomials the numbers in front of the x is there because you'll have like the zero power of x x the first x squared x cubed these numbers right here we call the coefficients of the polynomial and These are gonna be real numbers in our consideration here The let's see the the biggest term that shows up in this sum That is the power of x that has that the largest power of x This is commonly referred to as the leading term the coefficient of the leading term We call it the leading coefficient the exponent of the leading term We call this the degree of the polynomial and one thing we're gonna see very quickly in this Lecture is that of all the terms in a polynomial the leading term is the most important term Also, it's another another term in the polynomial that plays some role as can be the constant term over here Constant term because it doesn't have any variable x whatsoever the constant term Of course is going to determine the y-intercept of the graph The x to the first power we often refer to as the linear term X to the second power we might call the quadratic term x to the third We might call the cubic term and those are some names we've used for polynomials the linear functions We've studied in the past. These are examples of degree one polynomials the quadratic functions We studied in the previous chapter are what we call a degree two polynomials a Constant function is just a degree zero polynomial because it only has a constant term in this in this In this chapter, we'll of course study higher degree polynomials, right? So a degree three polynomial We might call that a cubic polynomial degree four known as a quartic Degree five might call this one a quintic And we could keep on going with vocabulary here, but we're not going to do much more of that But the polynomial is just a function which is a collection of all these different monomials If there's only one term in a polynomial, we could call that a monomial if there's two terms in the polynomial We might call it a binomial Being like a bicycle there's two if there's three terms in the polynomial We would often call that a trinomial kind of like a tricycle. There's more terms than that again There's vocabulary we could introduce but we probably aren't going to do it at that for that One thing that I can also mention about polynomials here is that that by the domain convention The domain convention told us originally that we want the domain to be as big as possible And we'll accept all real numbers for which the operations in play lead to well-defined real numbers when it comes to a polynomial The operations are going to be addition and subtraction. You're going to have exponents, which mean multiplication You have going to coefficients, which also mean multiplication. So when computing polynomials, you only need addition, subtraction, multiplication There is no restriction on those operations whatsoever So the domain of every single polynomial ever crafted is going to be negative infinity to infinity Finding the range is a little bit more difficult and that's a topic we will address later in this lecture