Understanding Polynomial Functions - Online Tutor, Practice Problems & Exam Prep

Polynomial functions are expressions with positive whole number exponents, written in standard form with descending powers. The degree indicates the highest power, while the leading coefficient is the numerical coefficient of that term. Understanding end behavior is crucial; it depends on the leading coefficient's sign and whether the degree is even or odd. Additionally, finding x-intercepts involves setting the polynomial equal to zero and factoring, considering the multiplicity of each zero to determine graph behavior at those points. The maximum number of turning points is given by the degree minus one, aiding in graph verification.

Introduction to Polynomial Functions

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 2. Now I want to take a broader look at all the possibilities of polynomial functions, which can really just be any polynomial, but now \( f(x) \) 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.

Looking at the polynomial function that I have right here, we want to remind ourselves of a couple of things that we 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 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 3, my next power is going to be 1 lower and then 1 lower, 1 lower until I get to the last one in 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 6, and \( n \) represents the degree of the polynomial. So in my polynomial here, it's just 3. Now when I look at my next term, I see that this power goes down to \( n-1 \), and that's just showing us that it's in descending order of power. So here I started with 3. If I subtract 1 from that, I get 2, which is my very next power here. And then all of these \( a \) terms down to my \( a_0 \) 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.

Okay, so let's just look at some polynomial functions here. So looking at this first one I have \( f(x) = -x^2 + 5x^3 - 6x + 4 \) 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, it 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 \( f(x) = 5x^3 - x^2 - 6x + 4 \), 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, my highest power, is simply 3. And then my leading coefficient is the one that's attached to that \( x^3 \). It is 5. So we're good on that one. Let's go ahead and move to our next one.

Here we have \( f(x) = 2x^{1/2} + 3 \). 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 anything else because it's not a polynomial function at all. Let's go ahead and move on to our last example here. We have \( f(x) = -\frac{2}{3}x^4 + 1 + 3 \). 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 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 \( f(x) = -\frac{2}{3}x^4 + 4 \), combining that 1 and 3. So let's identify our degree here. So my highest power on that first term is 4. And then lastly, my leading coefficient, looking here, I have -2/3 as my leading coefficient on that term with the highest power.

Okay, so we've taken a look at some polynomial functions, let's look at what their graphs may look like. So there are 2 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 know 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. It 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 okay. It is not a polynomial function because it has that break. So these 2 are not polynomial functions at all, and we want to look for things that look more similar to these 2 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 -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.

Determine if the given function is a polynomial function. If so, write in standard form, then state the degree and leading coefficient. $f\left(x\right)=4x^3+\frac12x^{-1}-2x+1$

• A. Polynomial with $n=3,a_{n}=4$
• B. Polynomial with $n=4,a_{n}=3$
• C. Polynomial with $n=-1,a_{n}=\frac12$
• D. Not a polynomial function.

Determine if the given function is a polynomial function. If so, write in standard form, then state the degree and leading coefficient. $f\left(x\right)=2+x$

• A. Polynomial with $n=1,a_{n}=2$
• B. Polynomial with $n=0,a_{n}=1$
• C. Polynomial with $n=1,a_{n}=1$
• D. Not a polynomial function.

End Behavior of Polynomial Functions

Hey, everyone. Whenever we worked with inequalities 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 4 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(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 f(x) goes far to the right side, 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 4 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 the standard form. So that's our leading term. And we're going to consider 2 things: the first, 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(x) will approach positive infinity, and if it is negative, the right side is instead going to fall, so f(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 on my 3rd example over here, it's going up. So it's rising. That means that my leading coefficient is positive. And then lastly, I am falling over here. That tells me that my leading coefficient is negative. Now a 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 5, 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, opposite if they're odd, they're going to be opposite. So look at that left side, here both sides are rising. They have the same behavior. That tells me that n, my degree, is even. Now, again, they're both falling, so they are again the same, so n is again even. Now, moving to my 3rd 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, f(x) is equal to -4x6+x3+2. 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 4. 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 6. 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, 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 with a parabola because we're not concerned with that yet.

So let's look at one final example here. Here we have f(x) is equal to 2x3+x. Now again, we only want to consider our leading term, our first term in standard form, which is already in standard form. So looking at that first term, my leading coefficient is a positive 2. 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 3, 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.

Determine the end behavior of the given polynomial function. $f\left(x\right)=x^2+4x+x+7x^3$

• 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

Match the given polynomial function to its graph based on end behavior. $f\left(x\right)=-2x^3+x^2+1$

• A.
• B.
• C.

Finding Zeros & Their Multiplicity

Hey, everyone. Whenever we worked with quadratic functions and their parabolas, we were able to find our x-intercepts by setting f(x) equal to 0 and most often getting either 1 or 2 x-intercepts by solving that quadratic equation. Now for polynomial functions, we could have 3 x-intercepts or even more than that. So how are we going to find all of them? Well, we're actually going to do the exact same thing and set f(x) equal to 0, 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 0, 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-12 that I use in order to get the zero x is equal to 1, my zero x equals 1 is going to have a multiplicity of 2 because that factor occurs twice. This is really just x minus 1 times x minus 1, and that's it. So let's go ahead and look at a more complicated polynomial here. So I have f(x)=2x×x-32×x+43. 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 0. So first, I have 2x is equal to 0, then I have this x minus 3 squared, which I know that squared isn't going to do anything, so I can just set x minus 3 equal to 0. And then I have x + 4 cubed, so I can set x + 4 equal to 0. So let's go ahead and do 2x is equal to 0 first. So if I divide both sides by 2, I'm simply left with x is equal to 0. Now over here with x minus 3, I can go ahead and add 3 to both sides, leaving me with x is equal to 3. Lastly, I have x + 4 equals 0. Subtracting 4 from both sides, I am left with x is equal to negative 4, and those are going to be my 3 zeros or my 3 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 0 that I got from 2x. Now, this factor only happens once, so it has a multiplicity of 1. Now looking at my second factor, for my second zero of x equals 3, which I got from the factor x minus 3 squared. Now this factor happens twice, so it has a multiplicity of 2. Now looking at my last zero, x is equal to negative 4, I got that from the factor x + 4 cubed, so this has a multiplicity of 3.

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 1, one 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 2, 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 3, 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.

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. $f\left(x\right)=2x^4-12x^3+18x^2$

• 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 $x=0$

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. $f\left(x\right)=x^2\left(x-1\right)^3\left(2x+6\right)$

• 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 $x=-3$

Maximum Turning Points of a Polynomial Function

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, 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 increasing 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. These points where our graph is changing direction are called turning points, where a graph is going from increasing to decreasing or decreasing to increasing, really just from going up to down and back up. So the maximum number of turning points that you can have is simply n − 1, where n is the degree of our polynomial.

Looking at our first example down here, I have 6x4 + 2x. So the degree of this polynomial is 4. If I simply take 4 and subtract 1, that gives me a maximum number of turning points of 3. Now we say maximum number because it doesn't have to have 3 turning points, but it just can have up to 3 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. This point right here has 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 valley, 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.

Looking at example b here, I have f(x) = x2 - 1. So the degree of this is simply 2. 2 - 1 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 -x2 + 5x3 - 6x. 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 3 right here. That is my degree. So 3 - 1 gives me a maximum number of turning points of 2, 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.

Determine the maximum number of turning points for the given polynomial function. $f\left(x\right)=6x^4+2x$

• A. 1
• B. 2
• C. 3
• D. 4

Based ONLY on the maximum number of turning points, which of the following graphs could NOT be the graph of the given function? $f\left(x\right)=x^3+1$

• A.
• B.
• C.

The given term represents the leading term of some polynomial function. Determine the end behavior and the maximum number of turning points. $4x^5$

• 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

Determine if the given function is a polynomial function. If so, write in standard form, then state the degree and leading coefficient. $f\left(x\right)=3x^2+5x+2$

• A. Polynomial with $n=3,a_{n}=2$
• B. Polynomial with $n=2,a_{n}=3$
• C. Polynomial with $n=2,a_{n}=2$
• D. Not a polynomial function.

Here’s what students ask on this topic:

What is a polynomial function and how is it different from a polynomial expression?

A polynomial function is a type of polynomial expression where the expression is set equal to a function, typically denoted as f(x). A polynomial expression consists of terms with positive whole number exponents, such as 3x² + 2x - 5. When this expression is set equal to f(x), it becomes a polynomial function, f(x) = 3x² + 2x - 5. The key difference is that a polynomial function represents a relationship between the input x and the output f(x), while a polynomial expression is just a sum of terms.

How do you determine the degree and leading coefficient of a polynomial function?

The degree of a polynomial function is the highest power of the variable x in the polynomial. For example, in the polynomial function f(x) = 5x³ - 2x² + 3, the degree is 3. The leading coefficient is the coefficient of the term with the highest power. In this case, the leading coefficient is 5. To find these, write the polynomial in standard form, which means arranging the terms in descending order of their powers.

What is the end behavior of a polynomial function and how is it determined?

The end behavior of a polynomial function describes how the function behaves as x approaches positive or negative infinity. It is determined by the leading term of the polynomial when written in standard form. If the leading coefficient is positive and the degree is even, both ends of the graph will rise. If the leading coefficient is negative and the degree is even, both ends will fall. If the degree is odd, the ends will do the opposite: one will rise and the other will fall, depending on the sign of the leading coefficient.

How do you find the x-intercepts of a polynomial function?

To find the x-intercepts of a polynomial function, set the function equal to zero and solve for x. This often involves factoring the polynomial. For example, for f(x) = 2x³ - 3x² - 5x + 6, set 2x³ - 3x² - 5x + 6 = 0 and factor it. The solutions to this equation are the x-intercepts. Each solution is called a zero of the function, and the multiplicity of each zero indicates how the graph behaves at that intercept.

What are turning points in a polynomial function and how are they determined?

Turning points in a polynomial function are points where the graph changes direction from increasing to decreasing or vice versa. The maximum number of turning points is given by the degree of the polynomial minus one. For example, a polynomial of degree 4 can have up to 3 turning points. These points can be local maxima or minima, depending on whether the graph is at the top of a hill or the bottom of a valley at those points.