Hello wonderful mathematics people. I'm Anna Cox from Kella Community College. In behavior of a graph, Basically, what happens to the Y when X goes to Infinity and when X goes to negative Infinity? If we start with some polynomial function, F of X equal a sub NX sub n + a sub n -, 1 X sub n - 1 plus... plus a sub One X + a sub zero. Our big important pieces here are the a sub N which is the leading coefficient and the N which is the exponent. It's the largest exponent of all of these terms being added and or subtracted for the leading coefficient test. If our a sub N is positive, then we know that if N is odd, if we have a really, really big number to an odd power, it's still going to be big. And if we're multiply it by some positive, we're going to get a big positive out towards Infinity. Now, we don't know what's happening in between, nor do we care, but we're going to look at what happens when we go out to negative Infinity. If I put a really, really, really big negative number in and it's to an odd power, a negative to an odd is going to be a negative. If we then multiply it by a positive coefficient, that's going to make a negative value when X goes out to negative Infinity for my Y. If we look at N is even, well, a big positive number to an even power is positive. And if you multiply it by a positive coefficient, it's still positive. I don't know what's going to happen in between if I take a big negative number to an even power, a big negative to an even power is positive, and positive times a positive coefficient is going to be a positive. So the black arrows represent what happens when X goes into Infinity and negative Infinity to our Y value. Now if the leading coefficient is negative, negative just really means a reflection through the X axis. So what's going to happen now is we're just going to flip the arrows. You could also think about it as a really really big positive number to an odd powers positive. But then if we multiply it by a negative coefficient, it turns into a negative value. I don't know what happens in between. All I'm discussing is what happens between as X goes to Infinity and negative Infinity. The last one really big positive number to an even power is going to be positive, but times a negative coefficient is going to make that Y value go to a negative. A really big negative number to an even power is positive. Times a negative coefficient would give me a negative. So the in behavior is what happens to the graph when we're going out to Infinity and negative Infinity doesn't tell me anything about what's occurring in between zeros of a polynomial function factor. The polynomial, then set each term equal to 0. So if we're given some polynomial to factor, when we have 4 terms, we're going to factor by grouping. We're going to group the 1st 2 together and the second two together. If I take out an X ^2 from the 1st 2 terms, I get X - 5. I'm going to take out a -4 from the 2nd 2 terms and get an X -, 5. Now these new 2 terms. So I went from 4 terms in the original to two terms. These two terms each have an X -, 5 in common. So using the distributive property, I'm going to pull it out. That leaves me an X ^2 - 4 X squared -4 is really just the difference of squares. So I get X -, 5 X -2 X +2. So the zeros of the polynomial would be X is five when Y is 0X is 2 when Y is 0X is -2 when Y is 0. With multiple or with zeros we also have multiplicities. And multiplicities just mean the degree of each of the terms. So if we look, multiplicity is the power of the factor or the exponent on each factored term. So when we look at the example we just did, the multiplicity on this 0 is going to be 1. The multiplicity on 20 is 1. The multiplicity on -2 zero is 1 because it's just the exponent and if there is no number, it's understood to be a one. If we had a new function, let's say F of X = -3 X cubed X -, 2 to the 4th, X + 7 to the 5th, and X - 3 to the first, the zeros would be 00, 20, negative 70 and 30. This was already as a monomial, it was already factored. It was already just multiplication of terms, the multiplicity of each term. The multiplicity of the 00 would be 3 because that's the exponent on the X. The multiplicity OF20 would be 4, of -7 zero would be 5, and of X -, 3 would be 1. Intermediate value theorem. If A is less than B and if F of A is less than 0 when F of B is greater than 0, then there exists some C such that A is less than C is less than B&F of C = 0. So pictorially, if AF of A is A is less than B&F of A is less than 0, and then BF of BF of B was greater than 0, What this says is somewhere we have to cross the axis. Now we may cross the axis multiple times. We don't know how many, we just know that we have to cross it at least once. So here would be our C Here we'd have lots of different values of CC1C2 and C3 would all be F of C = 0. Thank you and have a wonderful day. This is Anna Cox.