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Introduction to Polynomials - Concept

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

Before adding and subtracting polynomials or multiplying polynomials, it is important to have an introduction to polynomials with a definition of a polynomial and polynomial vocabulary. Important polynomial definitions include terms including monomial, the degree of a monomial, polynomial degree and standard form.

So polynomials are a new way of organizing a bunch of equations sorry new way of organizing a bunch of information it's basically a complicated equation. So what I have up here is the general equation for a polynomial and it looks pretty daunting but let's walk through it and start talking about what it means. So we basically have a bunch of x's and a bunch of a with these little subscripts and minus 1 and so on. But basically what that is saying is we have x to a power. Whatever that power is, it doesn't matter but that's a sub n is the subscript that matches with that. So say if this was 3x to the eighth, x to the eighth is the power and then this 3 is corresponding to a's of 8 because it's the coefficient on the term of the a power.
Now there are some conditions in what actually makes a polynomial and I have them written right here. First off all the a's have to be real numbers, basically they have to be numbers that we can put on a number line, they could be fractions, doesn't really matter as long as they are on the number line they exist. All the exponents have to be whole numbers so what that means is 1, 2, 3, 4, 5 so on and so forth even 0 is okay but they can't be negative, they can't be fractions anything like that.
And lastly we can't have any variables in the denominator so we can never divide by x and so on and so forth okay. Here example is another example of just a more specific one getting rid of all the n's and what have you. So this one right here has 4 terms okay all our coefficients are real numbers, all our exponents are whole numbers as well. This actually there is a term right here which is x to the 0 we don't need to write it because anything to the 0 is 1 so this actually disappears but it is really there. Okay some other lingo that goes along with this is degree and degree is how we talk about the powers of a polynomial and each term has a specific degree and not just the power on x okay. So this term right here is x to the fourth so it has the degree 4, x squared degree 2, x degree 1, x is 0 degree 0 okay. So each individual term has a degree but in addition the polynomial has a degree as well and that is just going to be the degree of the highest term.
Okay so this polynomial right here has a degree of 4, okay a common mistake is people want to add up all the degrees so this is degree this, this and this add them all together is not that complicated. Just look at the term of highest power in this case it's 4 that's our degree okay. In addition it tends to be convenient to write our polynomials in what's called descending order, it is basically from highest degree to lowest. But these polynomials are already in descending order because we started at a fourth degree, second, first, zero if this 4 we moved out in the front sorry if this 4 actually moved to the front it wouldn't be in descending order anymore because we'd have the first degree before the fourth, the descending order all our exponents are in down okay.
In addition there's a little bit more language that goes on with that, and basically you can have polynomials of different type okay. You can have polynomial as just a constant and what that mean is we don't have any degree okay so an example of that would just be f of x is equal to 2 okay there's an x to the 0 behind it but we don't need it because anything to the 0 is 1 so it's just a number, it could be 2, 7, negative 400 it doesn't really matter just a single term.
A linear term that's what we dealt with way back when lines okay there are no powers other than a single x so example of that 3x plus 1. The 3 in the 1 are arbitrary but we just the main thing is having a single x to the first power degree 1. Quadratic, quadratic is a degree 2 okay, so to have these polynomials we actually don't need all these terms so we could just say example of a quadratic is just x squared. Okay it's a very simple quadratic or we could also say x squared plus 4x minus 2 doesn't really matter as long as we have that x squared term and only degrees less than that, it would be a quadratic.
Lastly cubic you guessed it degree 3 is our highest degree, let's say negative 4x cubed and whatever is here doesn't matter as long as the degree is 2, 1, and 0 okay. For general introduction to polynomials sort of the abstract long equation and some rules we have and how we talked about them.

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