Understanding Polynomial Functions - Video Tutorials & Practice Problems
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1
concept
Introduction to Polynomial Functions
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6m
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Hey, everyone, we've worked with polynomial expressions and we've even worked with a specific type of polynomial function. A quadratic function where my highest power is too. Now, we want to take a broader look at all the possibilities of polynomial functions, which can really just be any polynomial. But now f of X is equal to that polynomial making it a polynomial function. So that might sound like a lot, any polynomial. And now it can be a function too. But don't worry, we're going to rely on a lot of what we already know about polynomials and some of what we just learned about quadratic functions in order to learn more about polynomial functions and their graphs. So let's go ahead and jump right in. So looking at the polynomial function that I have right here, we want to remind ourselves of a couple of things that we've learned with polynomials. The first of which is that polynomials can only have positive whole number exponents. So that means no negatives and no fractions in those exponents. The other thing is that whenever we write our polynomials in standard form, all of our like terms need to be combined and it needs to be written in descending order of power. So if I start with a power of three, my next power is going to be one lower and then one lower, one lower until I get to the last one descending order. Now, looking at these polynomials that I have here, you might notice this one looks similar to what you've seen in your textbook and this can look a little bit intimidating when you first see it. But don't worry, this is just showing us exactly how to write any polynomial in standard form. So this A N right here represents my leading coefficient. So in the polynomial that I have here, it's simply six and N represents the degree of the polynomial. So in my polynomial here, it's just three. Now, when I look at my next term, I see that, that this power goes down to N minus one and that's just showing us that it's in descending order of power. So here I started with three. If I subtract one from that I get two, which is my very next power here. And then all of these a terms down to my, a subzero just represent all of my coefficients and then my constant. So don't let that form freak you out. It's just showing you how to write in standard form. OK? So let's just look at some polynomial functions here. So looking at this first one I have F of X is equal to negative X squared plus five X cubed minus six X plus four. And I want to determine if this even is a polynomial function. And if it is write it in standard form and state both the degree and leading coefficient. So let's first determine if this even is a polynomial function. And since I only have positive whole number exponents, I don't have any fractions or any negatives in those exponents looks like yes, this is a polynomial function. So let's go ahead and write it in standard form. Now here it looks like I just need to switch these two terms in order for it to be in standard form. So I end up with F of X is equal to five X cubed minus X squared minus six X plus four. And we're good, we're in standard form. So let's go ahead and identify that degree. So the degree of this polynomial looking at that first term I is power is simply three. And then my leading coefficient is the one that's attached to that X cubed. It is five. So we're good on that one. Let's go ahead and move to our next one. Here. We have F of X is equal to two X to the power of one half plus three. Now, is this a polynomial function? Well, I have a fraction in my exponent there. So no, this is not a polynomial function. I cannot have fractions as exponents. So I don't have to worry about any else because it's not a polynomial function at all. Let's go ahead and move on to our last example here we have F of X is equal to negative two thirds times X to the fourth power plus one plus three. Now, looking at this, there is a fraction as a coefficient. Does that mean that it's not a polynomial? Well, no, because it's just a coefficient. It's not in my exponent and my exponents here are positive whole numbers. So it looks like yes, this is, is a polynomial function. You can have a fraction as a coefficient, just not as an exponent. So let's go ahead and write this in standard form which I can do by simply combining these like terms that I have here. So I end up with F of X is equal to negative two thirds X to the fourth plus four, combining that one and three. So let's identify our degree here. So my highest power on that first term is +84. And then lastly my leading coefficient looking here, I have negative two thirds as my leading coefficient on that term with the highest power. OK. So we've taken a look at some polynomial functions. Let's look at what their graphs may look like. So there are two things that we want to think about with the graphs of polynomial functions and that is that they are both smooth and continuous. This will be true. Of the graph of any polynomial function. And what that means is that there will be no corners in our graph and there will be no breaks. So looking at this, these polynomial functions on the left side, I have this curve here that is smooth, it is a smooth curve, it is continuous, it never breaks off. And then here you might recognize this as a quadratic function which we no is definitely a polynomial function and it is both smooth and continuous as well. Now, looking over to this right side here and this graph, I have a really harsh corner right here. So that tells me that I am not dealing with a polynomial function is not a polynomial function at all. It is not smooth and continuous. So let's look at one more here and looking at this graph, it breaks off and then it keeps going on that side. Now this is not OK. It is not a polynomial function because it has that break. So these two are not polynomial functions at all. And we want to look for things that look more similar to these two that are both smooth and continuous. So one more thing that I want to mention about the graphs of polynomial functions is the domain which is always for any polynomial function going to go from negative infinity to infinity, which is something you may remember from working with quadratic functions. All quadratic functions had this domain. And all polynomial functions have it as well. All real numbers are included in that domain. So that's a little of the basics of polynomial functions. Thanks for watching and I'll see you in the next video.
2
Problem
Problem
Determine if the given function is a polynomial function. If so, write in standard form, then state the degree and leading coefficient. f(x)=4x3+21x−1−2x+1
A
Polynomial with n=3,an=4
B
Polynomial with n=4,an=3
C
Polynomial with n=−1,an=21
D
Not a polynomial function.
3
Problem
Problem
Determine if the given function is a polynomial function. If so, write in standard form, then state the degree and leading coefficient. f(x)=2+x
A
Polynomial with n=1,an=2
B
Polynomial with n=0,an=1
C
Polynomial with n=1,an=1
D
Not a polynomial function.
4
Problem
Problem
Determine if the given function is a polynomial function. If so, write in standard form, then state the degree and leading coefficient. f(x)=3x2+5x+2
A
Polynomial with n=3,an=2
B
Polynomial with n=2,an=3
C
Polynomial with n=2,an=2
D
Not a polynomial function.
5
concept
End Behavior of Polynomial Functions
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6m
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Hey, everyone, whenever we worked with any qualities and we graphed them, we saw that our graph could continue on to infinity, whether it be positive infinity or negative infinity. Now, the graphs of polynomial functions are going to do the same thing continue on to infinity, whether it be positive or negative. But we're no longer working with simple inequalities. So how will we determine what the graph of our polynomial is doing? Going to a positive or negative infinity? Well, it might sound like it's going to be complicated, but I'm going to show you how it's actually just going to be four possible things that could be happening on the ends. And we only need to look at one single term in our polynomial function to determine this. So let's go ahead and jump in. Now, this behavior on either end of our graph is very creatively named the end behavior. And it's simply referring to what F of X our graph is doing far to the left side. So looking as our graph goes far to the left, that's as X approaches negative infinity in that arrow notation or as X goes far or as F of X goes far to the right side. So going all the way to positive infinity in that arrow notation as X approaches positive infinity. So let's go ahead and look at the four possible things that could be happening. Now, looking at each of these, the behavior in the middle of the graph is going to be different based on the function. So here we're really only concerned with what's happening on the ends and something different could be happening in the middle again, depending on your function, waving up and down or doing a number of things. But let's just focus on our end behavior. Now, like I said, there's only one term in our polynomial function that we need to consider. And that is going to be the first term when our function is written in standard form. So that's our leading term. And we're going to consider two things. The first of which is going to be the leading coefficient of our polynomial function. Now, the leading coefficient we're going to look at the sign and if the sign is positive, then the right side of our graph is going to rise. So F of X will approach positive infinity and if it is negative, the right side isn't instead going to fall. So F of X will approach negative infinity. So let's look at our graphs up here and determine whether we're dealing with a positive leading coefficient or a negative leading coefficient. So on my first one looking at that right side, my right side is rising. That tells me that my leading coefficient is positive. Now, over here, I have my right side falling down. So that tells me that my leading coefficient is instead negative. Now my third example over here it's going up. So it's a rising. That means that my leading coefficient A N is positive. And then lastly, I am falling over here, that tells me that my leading coefficient is negative. Now, way that I like to think about this is if my leading coefficient is positive, that's good. It's going to rise up. And if my leading coefficient is negative, that's bad, it's going to fall back down. Now that we've looked at that right side, what's happening on the left side. So the other thing that we're going to look at in that first term is going to be the degree of our polynomial and whether it is even or odd. So if that degree is even, then the ends are going to be the same. So that means that if one side is rising, the other one has to be rising as well. And if they are odd, the ends are instead going to do the opposite thing have the opposite behavior. So in my polynomial right here, if my degree is five, that's an odd number that tells me that my ends are going to do the opposite thing. Now, something that I like to use to remember this is odd, opposite kind of sounds a little bit similar. Odd. If they're odd, they're going to be the opposite. So let's go ahead and go back to our examples here. So looking at now, at that left side here, both sides are rising. They have the same behavior that tells me that in my degree is even now, here again, they're both falling. So they are again the same. So N is again, even now moving to my third possibility, they are opposite. One side is rising while the other is falling. So odd opposite N is odd. And then lastly, my right side over here is falling, my left side is rising again, opposite odd opposite. So my degree is odd. So those are my four possibilities. Let's go ahead and look at some polynomials and try to sketch their end behavior. So looking at this first example here, ff of X is equal to negative four X to the power of six plus X cubed plus two. So looking at this, I only want to consider the first term. So the first thing I'm going to look at is my leading coefficient, which here is negative four. Now that leading coefficient is negative, that's bad. It's going to fall down on that right side. And then the other thing I want to look at here is the degree. So my degree here is six, that is an even number. So it is not odd opposite is even and the same. So my ends are going to have the same behavior. Now to sketch this, I'm going to consider that my right side is falling, go ahead and indicate that. And then if my ends are the same and my left side has to be falling as well. Now, remember we're not focused on what's happening in the middle. So you could really connect them in any way. But here, I'm just going to connect them the parable up because we're not concerned with that yet. So let's look at one final example. Here, here we have F of X is equal to two X cubed plus X. Now again, we only want to consider our leading term, our first term in standard form, which it's already in standard form. So looking at that first term, my leading coefficient is a positive two. So since it is positive, that's good that my right side is going to rise up and then looking at my degree, my degree is three, which is an odd number odd opposite. So my ends are going to have the opposite behavior. So let's go ahead and sketch this. So again, my right side is going to rise, let's go ahead and draw that it's rising up and then my ends are the opposite. So I know that the left side has to do the opposite and fall back down. And I'm just gonna connect this in the middle remember, we're not yet concerned with that. So that's all you need to know about the end behavior of a polynomial. Let's go ahead and get some practice.
6
Problem
Problem
Determine the end behavior of the given polynomial function. f(x)=x2+4x+x+7x3
A
Right side rises; Ends are same
B
Right side rises; Ends are opposite
C
Right side falls; Ends are same
D
Right side falls; Ends are opposite
7
Problem
Problem
Match the given polynomial function to its graph based on end behavior. f(x)=−2x3+x2+1
A
B
C
8
concept
Finding Zeros & Their Multiplicity
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3m
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Hey, everyone, whenever we worked with quadratic functions and their parabolas, we were able to find our X intercepts by setting F of X equal to zero. And most often getting either one or two X intercepts by solving that quadratic equation. Now, for polynomial functions, we could have three X intercepts or even more than that. So how are we going to find all of them? We're actually going to do the exact same thing and set F of X equal to zero. But the problem still remains, how do I solve that for some complicated polynomial? And it's actually something that we've done a million times before. And that is simply by factoring. So really, we're just going to factor set each factor equal to zero. And then we're going to look at something called the multiplicity of each zero. So I'm going to walk you through all of this. Let's go ahead and get started. So I mentioned the multiplicity of the zero and this is really just the number of times that a factor occurs. So if I have a factor X minus one squared that I use in order to get the zero X is equal to one. My zero X equals one is going to have a multiplicity of two because that factor occurs twice. This is really just X minus one times X minus one and that's it. So let's go ahead and look at a more complicated polynomial here. So I have F of X is equal to two X times X minus three squared times X plus four cubed. Let's go ahead and solve for each zero. This is already factored for me. So I can go ahead and just take each individual factor and set them equal to zero. So first, I have two, X is equal to zero, then I have this X minus three squared which I know that squared isn't going to do anything. So I can just set X minus three equals to zero. And then I have X plus four cube so I can set X plus four equal to zero. So let's go ahead and do two. X is equal to zero first. So if I divide both sides by two, I'm simply left with X is equal to zero. Now, over here with X minus three, I can go ahead and add three to both sides leaving me with X is equal to three. And lastly, I have X plus four equals zero. Subtracting four from both sides. I am left with X is equal to negative four and those are going to be my three zeros or my three X intercepts. Let's go ahead and identify the multiplicity of each of these zeros as well. So looking at this first factor or my first zero of X equals zero that I got from two X. Now this factor only happens once. So it has a multiplicity of one. Now looking at my second factor, my second zero X equals three, which I got from the factor X minus three squared. Now this factor happens twice. So it has a multiplicity of two. Now looking at my last zero X is equal to negative four, I got that from the factor X plus four cubed. So this has a multiplicity of three. Now, what's the point of multiplicity? Why do we even need to find it? Well, multiplicity is actually going to tell us the behavior of our graph at each zero. So if our multiplicity is even that tells us that our graph is going to touch our X axis and simply bounce right back off. So at this point here, it just touches our X axis bounces back off in the same direction it came from. Now, if our multiplicity is instead odd, our graph is going to fully cross our X axis going from one side to the other, like we see at these two points. So let's go ahead and identify that in our example. So if I had this multiplicity of 11 is an odd number, so that means that my graph is going to fully cross the X axis at that zero. Now, here I have a multiplicity of two, which is an even number. So it's simply going to touch the X axis and bounce right back off. Lastly, here I have a multiplicity of three, which is again an odd number. So my graph is going to fully cross my X axis again at that point. So that's all you need to know about zeros and their multiplicity. Let's get some practice.
9
Problem
Problem
Find the zeros of the given polynomial function and give the multiplicity of each. State whether the graph crosses or touches the x-axis at each zero. f(x)=2x4−12x3+18x2
A
Touch at x=0, Cross at x=−3
B
Touch at x=0, Touch at x=3
C
Touch at x=1, Cross at x=−3
D
Touch at x=−1, Cross at x=0
10
Problem
Problem
Find the zeros of the given polynomial function and give the multiplicity of each. State whether the graph crosses or touches the x-axis at each zero. f(x)=x2(x−1)3(2x+6)
A
Cross at x=0, Cross at x=1, Cross at x=3
B
Touch at x=0, Cross at x=−1, Cross at x=3
C
Cross at x=0, Touch at x=1, Touch at x=−3
D
Touch at x=0, Cross at x=1, Cross at x=−3
11
concept
Maximum Turning Points of a Polynomial Function
Video duration:
2m
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Hey, everyone, we've been looking at the different elements of the graphs of polynomial functions like the end behavior as X approaches negative infinity or positive infinity and the zeros of our function and what our graph is doing at each of those zeros. So whether it is crossing our X axis like at this point or simply touching and bouncing off like at this point. So we have left to consider what's going on in between all of these points as our graph is going from increase to decreasing or decreasing to increasing. And how many times it can do that. So I'm going to show you exactly how to determine how many times our graph can change direction using one simple thing, the degree of our polynomial. So let's go ahead and jump right in. So these points where our graph is changing direction are called turning points. So where a graph is going from increasing to decreasing or decreasing to increasing, really just from going up to down to down and back up. So the maximum number of turning points that you can have is simply N minus one where N is the degree of our polynomial So looking at our first example down here, I have six X to the fourth power plus two X. So the degree of this polynomial is four, if I simply take four and subtract one, that gives me a maximum number of turning points of three, we say maximum number because it doesn't have to have three turning points, but it just can have up to three turning points. One other thing that we want to consider with our turning points is that they will either be a local maximum or a local minimum depending on what direction is it is changing from. So this point right here has a lower points. On either side, it is at the top of a hill, it represents a maximum. Whereas this point down here it is going up on either side, it's in a little value. So it is a minimum point. Now, that's just something to consider about turning points. But let's go ahead and calculate some more maximum numbers of a couple more polynomial functions. So looking at example B here, if of X is equal to X squared minus one. So the degree of this is simply 22 minus one, gives me one maximum turning point, which makes sense because I know that the shape of this graph is like this only one turning point because it is a quadratic function. Let's look at one final example here. So I have negative X squared plus five X cubed minus six X. Now, remember the degree is our highest power and this is not in standard form. So it's not my first term, but I have this three right here. That is my degree. So three minus one, it gives me a maximum number of turning points of two and that's all. So you might be wondering what we're going to use turning points for, why would I need to know the maximum number of turning points? And it's really just going to serve as a way to check that we have graphed a polynomial function correctly. So that's all you need to know about turning points. Let's go ahead and get to graphing.
12
Problem
Problem
Determine the maximum number of turning points for the given polynomial function. f(x)=6x4+2x
A
1
B
2
C
3
D
4
13
Problem
Problem
Based ONLY on the maximum number of turning points, which of the following graphs could NOT be the graph of the given function? f(x)=x3+1
A
B
C
14
Problem
Problem
The given term represents the leading term of some polynomial function. Determine the end behavior and the maximum number of turning points. 4x5
A
Right side rises; Ends are opposite & 4 maximum turning points
B
Right side rises; Ends are opposite & 5 maximum turning points
C
Right side rises; Ends are the same & 4 maximum turning points
D
Right side falls; Ends are opposite & 4 maximum turning points