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INSTRUCTORSCarleen EatonGrant Fraser
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Polynomial Functions

  • A polynomial function can be evaluated for an algebraic expression.
  • Graphs of polynomial functions of degree n suggest that the maximum number of real zeros of such a function is n.
  • The end behavior of a polynomial function depends on the sign of the leading coefficient and whether its degree is even or odd.

Polynomial Functions

What is the degree and leading coefficient of 5x − 11x2 + 25 − x3 − 7x4?
  • Write the polynomial in standard form, in other words, descending order.
  • − 7x4 − x3 − 11x2 + 5x + 25
  • Degree − 4
Leading Coefficient: − 7
What is the degree and leading coefficient of x − x2 + 25 + 10x5?
  • Write the polynomial in standard form, in other words, descending order.
  • 10x5 − x2 + x + 25
  • Degree: 5
Leading Coefficient: 10
What is the degree and leading coefficient of − 5x + x3 + 2 − 5x7?
  • Write the polynomial in standard form, in other words, descending order.
  • − 5x7 + x3 − 5x + 2
  • Degree: 7
Leading Coefficient: − 5
Let f(x) = 2x3 − x2 + x − 1 find f(2a2)
  • Substitute 2a2 everywhere you see an x
  • f(2a2) = 2(2a2)3 − (2a2)2 + (2a2) − 1
  • Simplify using Power of Power Rule of Exponents
  • f(2a2) = 2(23a6) − (22a4) + (2a2) − 1
  • Simplify
f(2a2) = 2(8a6) − (4a4) + (2a2) − 1 = 16a6 − 4a4 + 2a2 − 1
Let f(x) = x3 + x2 + x + 1 find f( − a3)
  • Substitute − a3 everywhere you see an x
  • f( − a3) = ( − a3)3 + ( − a3)2 + ( − a3) + 1
  • Simplify using Power of Power Rule of Exponents.
  • To better work with the negative and exponents, subsitute a − 1 to make arithmetic easy to work with.
  • Warning - You may not factor out a negative out, that will lead to wrong results.
  • f( − a3) = ( − 1*a3)3 + ( − 1*a3)2 + ( − a3) + 1 = ( − 1)3a9 + ( − 1)2a6 − a3 + 1
  • Simplify. Notice how a negative number raised to an odd power is always negative.
  • On the other hand, a negative number raised to an even power is always positive.
  • f( − a3) = ( − 1)3a9 + ( − 1)2a6 − a3 + 1 = − 1*a9 + 1*a6 − a3 + 1
f( − a3) = − a9 + a6 − a3 + 1
Let f(x) = x5 + x3 + x find f( − a2)
  • Substitute − a2 everywhere you see an x
  • f( − a2) = ( − a3)5 + ( − a2)3 + ( − a3)
  • Simplify using Power of Power Rule of Exponents.
  • To better work with the negative and exponents, subsitute a − 1 to make arithmetic easy to work with.
  • Warning - You may not factor out a negative out, that will lead to wrong results.
  • f( − a2) = ( − a3)5 + ( − a2)3 + ( − a3) = ( − 1*a3)5 + ( − 1*a2)3 + ( − a3) = ( − 1)5a15 + ( − 1)3a6 − a3
  • Simplify. Notice how a negative number raised to an odd power is always negative.
  • On the other hand, a negative number raised to an even power is always positive.
  • f( − a2) = ( − 1)5a15 + ( − 1)3a6 − a3 = − 1*a15 − 1*a6 − a3
f( − a2) = − a15 − a6 − a2
Let f(x) = x2 + x + 5. Find f(2b − 1)
  • Substitute 2b − 1 everywhere you see an x.
  • f(2b − 1) = (2b − 1)2 + (2b − 1) + 5
  • Be careful, you cannot use the properties of exponents in this step. In other words, you cannot do
  • (2b − 1)2 = 22b2 − 12. Doing this will lead to wrong results.
  • Multiply using FOIL, or anyother method of your preference.
  • f(2b − 1) = (2b − 1)2 + (2b − 1) + 5
  • = (2b − 1)(2b − 1) + (2b − 1) + 5
  • = 4b2 − 4b + 1 + (2b − 1) + 5
  • Combine like terms
f(2b − 1) = 4b2 − 2b + 6
Let f(x) = 2x2 − 3x + 7. Find f(3b + 2)
  • Substitute 3b + 2 everywhere you see an x.
  • f(3b + 2) = 2(3b + 2)2 − 3(3b + 2) + 7
  • Be careful, you cannot use the properties of exponents in this step. In other words, you cannot do
  • (3b + 2)2 = 32b2 + 22. Doing this will lead to wrong results.
  • Multiply using FOIL, or anyother method of your preference.
  • f(3b + 2) = 2(3b + 2)2 − 3(3b + 2) + 7
  • = 2(3b + 2)(3b + 2) − 3(3b + 2) + 7
  • = (6b + 4)(3b + 2) − 9b − 6 + 7
  • = 18b2 + 24b + 8 − 9b − 6 + 7
  • Combine like terms
f(2b − 1) = 18b2 + 15b + 9
Let f(x) = − 2x6 − x4 + 2x2 − 5x + 1. Describe the end behavior.
  • To answer this question, you need two pieces of information and the following table.
  • 1)Degree of the polynomial
  • 2)Sign of the leading coefficient
  • End Behavior Chart Even Degree Odd Degree
    + leading coefficient 1. As x approaches +∞, y approaches +∞ 1. As x approaches +∞, y approaches +∞
      2. As x approaches -∞, y approaches +∞ 2. As x approaches -∞, y approaches -∞
    - leading coefficient 1. As x approaches +∞, y approaches -∞ 1. As x approaches +∞, y approaches -∞
      2. As x approaches -∞, y approaches -∞ 2. As x approaches -∞, y approaches +∞
  • In this problem, the Degree is 6 and the leading coefficient is negative. Therefore:
As x gets really big approaching positive infinity,y is going to get really small approaching negative infinity.

As x gets really small approaching negative infinity, y is going to get really small approaching negative infinity.
Let f(x) = 2x9 + x5 + 2x3 − 5x2 + x − 10. Describe the end behavior.
  • To answer this question, you need two pieces of information and the following table.
  • 1)Degree of the polynomial
  • 2)Sign of the leading coefficient
  • End Behavior Chart Even Degree Odd Degree
    + leading coefficient 1. As x approaches +∞, y approaches +∞ 1. As x approaches +∞, y approaches +∞
      2. As x approaches -∞, y approaches +∞ 2. As x approaches -∞, y approaches -∞
    - leading coefficient 1. As x approaches +∞, y approaches -∞ 1. As x approaches +∞, y approaches -∞
      2. As x approaches -∞, y approaches -∞ 2. As x approaches -∞, y approaches +∞
  • In this problem, the Degree is 9 (odd) and the leading coefficient is 2 (positive). Therefore:
As x gets really big approaching positive infinity, y is going to get really big approaching positive infinity.

As x gets really small approaching negative infinity, y is going to get really small approaching negative infinity.

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Polynomial Functions

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Polynomial in One Variable 0:13
    • Leading Coefficient
    • Example: Polynomial
    • Degree
  • Polynomial Functions 2:57
    • Example: Function
  • Function Values 3:33
    • Example: Numerical Values
    • Example: Algebraic Expressions
  • Zeros of Polynomial Functions 5:50
    • Odd Degree
    • Even Degree
  • End Behavior 8:28
    • Even Degrees
    • Example: Leading Coefficient +/-
    • Odd Degrees
    • Example: Leading Coefficient +/-
  • Example 1: Degree and Leading Coefficient 15:03
  • Example 2: Polynomial Function 15:56
  • Example 3: Polynomial Function 17:34
  • Example 4: End Behavior 19:53

Transcription: Polynomial Functions

Welcome to Educator.com.0000

Today, we are going to talk about polynomial functions, starting with some review,0002

and then going on to discuss the topic of analyzing the graphs of polynomial functions.0006

OK, a polynomial in one variable is what we are going to start out with; and a polynomial of the degree n,0013

in one variable x, is an expression in this following form.0021

an, the leading coefficient, cannot be 0, because if it was to be 0, this would drop out;0028

and then you would actually have a lower-degree polynomial.0033

And n is a non-negative number; these are not going to have a negative coefficient up here.0036

This first coefficient is called the leading coefficient.0046

A polynomial is just an expression in which the terms are monomials; and a restriction on this is that there cannot be variables in the denominator.0052

If there were variables, it would not be in this form; if that were the case, you would have something else.0062

You would have a rational expression, not a polynomial.0065

Now, let's talk about degree, using numbers instead of just n.0070

If you have something such as 3x4 (a polynomial) + x3 - 4x2 - x + 4,0074

the degree of the polynomial is the degree of the variable that is the highest power.0092

So, right here, the highest power I have is 4; therefore, the degree equals 4.0100

And the leading coefficient is the coefficient for the variable that is raised to the highest power.0109

One thing to be aware of is that you might look at a polynomial, and it could be written like this:0115

2x2 - 6x5 + 4x3 + 9.0121

And when you look at this, you have to realize that these are not written in descending order.0131

So, you can't just look at the first one and say, "Oh, this is degree 2."0136

Before you figure out what the degree of a polynomial is, or what the leading coefficient is, you first need to write it in descending order.0140

So, that would be -6x5 + 4x3 + 2x2 + 9.0148

Here, the degree is actually 5, and the leading coefficient is -6.0156

Just make sure that the terms are in descending order before you determine the degree and the leading coefficient.0170

Now, polynomial functions: we just saw this form, but now we are talking about it as a function.0177

A polynomial function of degree n is a function in this form.0184

You may see this written as f(x), like this; you may also see p(x); that is sometimes used for a polynomial function.0189

And just to give you an example, I am going to call it f(x) (but it could have been called p(x)) = 2x3 - 4x2 + 8x - 5.0199

This is a polynomial function; and with polynomial functions, we can evaluate the function for both numerical values and algebraic expressions.0210

You are probably familiar with evaluating functions for numerical values.0220

But evaluating them for algebraic expressions may be something new.0227

So, first, just looking at a function: f(x) = 4x3 - 3x2 + x - 1: if I am asked to find f(2),0231

the strategy is just going to be to replace all of these x's with 2's.0244

which is going to give me...2 times 2 is 4, times 2 is 8; so it is 4 times 8 (23 is 8) minus 3 times 2 (is 6), plus 2, minus 1.0256

This is 32 minus...actually, correction: this is an extra place with 2...2 times 2 is 4, because that is squared; so it is going to give me 3 times 4.0267

So, I am going to end up with 4 times 8, minus 3 times 4, plus 2, minus 1.0283

This is going to give me 32 - 12 + 2 - 1; simplifying that, 32 - 12 is 20, plus 2 is 22, minus 1--that is 21, so f(2) is 21.0289

That is straightforward, but something you may be less familiar with is the idea of something like this: f(a + 3).0306

Again, I am going to substitute this expression for the x's--the same idea, only now I have a variable in here.0316

Wherever there is an x, I am going to put in a + 3 instead.0326

Now, you could do the algebra and work this out to figure out the value; but the important part0333

is just knowing how to set this up by substituting this algebraic expression for the variable in the function.0338

OK, a point at which the graph of a function intersects with the x-axis is called a zero of the function.0348

And previously, we have referred to x-intercepts: this is the same thing, only a different term.0355

And the degree of the function can tell you something about the zeroes.0363

If a polynomial function has an odd degree, there is at least one zero.0370

For example, if I have something like f(x) = 6x3 - 5x2 + 8x, the degree here is 3, and it is odd.0374

So, I know that it has at least one zero.0388

Now, just showing you examples of zeroes: we talked about linear functions, which graph as a line,0392

and then quadratic functions (which are polynomial functions) are parabolas; you can also have0402

more elaborate shapes, such as this, with a polynomial.0407

And everywhere that intersects with the x-axis, those are all zeroes; so this actually has 1, 2, 3, 4, 5 zeroes.0411

And let's say this is 2, 4, 6, 8, 10; -2, -4, -6, -8, -10; you could find the values for these.0423

For example, one of the zeroes is at 4; so you could find the values for these zeroes.0435

OK, so you always have at least one zero, if the degree of the polynomial function is odd.0443

If the degree is even, it may or may not have a zero (it may or may not intersect with the x-axis).0449

So, something like this might have a zero and might not.0456

The degree equals 4; it is even; it may or may not have a zero.0466

Thinking back to quadratic equations (their degree is 2--something like x2 + 2x - 1--quadratic functions),0473

recall that these parabolas sometimes have a zero; they sometimes intersect with the x-axis.0481

They can have one or two zeroes (this one has two), or they may not intersect.0487

We saw situations where parabolas did not intersect, and those have even degrees.0493

And the same is true of a polynomial function with a greater degree that is even.0498

When we talk about functions, we sometimes talk about end behavior.0510

And the end behavior of a function refers to what the graph does as x gets very large or very small.0514

So, instead of in the middle (which is often what we have been focusing on with graphing), now we are thinking about the ends--0520

way out here when x is very positive, or way out here when x is very negative.0527

And you can predict some of this end behavior by looking at the degree of the polynomial function and the sign of its leading coefficient.0533

So, let's break this down into first talking about even degrees.0545

And then, we will talk about odd degrees; polynomials of even degrees first.0554

For example, x4 - 2x3 + x - 8: this has a degree of 4--it is even.0563

We can divide this into two categories: those that have a leading coefficient that is positive...0577

if the leading coefficient, an, is greater than 0, both ends of the graph go upward.0589

Now, let's think about what that means.0605

For example, if I have some polynomial function, and it is even (so it may or may not have a zero,0607

but I am going to show it as having zeroes), both ends are up; so this is even,0612

and an is greater than 0; both of these ends go up.0622

What this is saying is that, when x is very large, over here, or x is very small, way out here, f(x) is large.0626

So, end behavior--what is happening way out here at the extremes--when x is very large0648

or when x is very small, you see that y is going to go up, up, up and be very large.0653

So, that is an even degree; both ends are up; and it is a positive leading coefficient,0659

so both ends are up; there are very large y's, or function values, at the extreme values of x.0665

Now, the other possibility (that is one possibility for even) is that the leading coefficient is less than 0.0672

I have a negative leading coefficient--something like -4x2 + 3x - 1; and this is actually a quadratic function.0680

And if you recall, this is going to be a parabola that faces downward.0693

Let's look at this: here, this could have had a higher even degree, like 4 or 6 or 8; but let's just look at the parabola.0700

Here, both ends face downward; and what this tells me is that, when x is large, or when x is small0711

(very small or very large), f(x) is small--it has small values.0734

So, at the ends here, when x is these small values or very large values, the y is going to have very small values.0742

OK, so these are even degrees; and to help you remember this, just think back to quadratic equations,0754

because we already talked about quadratic functions and the fact that, if their leading coefficient is positive, they face upward.0758

If their leading coefficient is negative, they face downward.0764

And it is the same idea here, except you might have more ups and downs in between.0767

Now, if you have a polynomial with an odd degree, again, we have two possibilities.0771

If I have a polynomial that has an odd degree, and the leading coefficient is positive (a is greater than 0),0779

then what I am going to have is a graph that is going to go up to the right; that is this case right here.0791

What this is saying is that, when x is large, y is large (or f(x) is large--either way you want to say it).0808

And when x is small, y gets very small.0822

So, I just think of it as "x and y are going in the same directions."0830

Now, if the leading coefficient is negative, we have the opposite situation; here you would have something like this, where...0833

this is odd degree, and it is an a that is negative; here I see that it is starting up here, and it is going down to the right.0845

The graph goes down to the right.0864

So, with even degrees, both ends face up, or both ends face down; with odd degrees, one end goes up; the other goes down.0870

If it is a positive leading coefficient, it goes up to the right; if it is a negative leading coefficient, it goes down to the right.0879

And you also saw that with linear equations--going up or down to the right.0885

What this tells me is that, when x is very large, y is very small; and when x is very small, y is very large.0891

x and y are going the opposite ways.0898

OK, first looking at this polynomial: what is the degree and leading coefficient?0903

I am going to start out by putting this in descending order; this is my largest degree here, so look at that first.0909

Then, I have x3; I have -7x2; 3x; and finally, the constant.0917

OK, so now that I have this in descending order, I can easily see that the degree equals 4.0925

And the leading coefficient is going to be the coefficient for that variable, raised to the largest power.0932

The leading coefficient is -12; and if I didn't put these in the right order, and just looked at it, I might have said, "Oh, the leading coefficient is 3."0941

So, always put your polynomial in descending order first.0951

OK, given this function, find f(3a4).0957

This is an algebraic term, and we need to find the function value for it; so I am going to substitute 3a4 everywhere there is an x.0965

I am going to use my rule here of raising a power to a power; that tells me that, when I raise a power to a power, I need to multiply the exponents.0980

So, first, I have my 3 here; let's keep that out here; and I have to remember to raise 3 to 3, also.0988

So, this is 3 times 3 is 9, times 3 is 27; so that is 27a; and that is 4 times 3, so that is a12.0998

And then here, I have 32, which is 9; a4...OK, that is 4 times 2, which is 8.1010

And then, this just stays like this.1020

Next, I have 3 times 27, which is actually 81; a12; -2 times 9 gives me -18a8;1023

plus 3a4, minus 7; and I always double-check and make sure I can't do any further simplification.1035

And I can't; I don't have any like terms that I can combine.1042

So again, evaluating a function for an algebraic expression just involves substituting that expression in for the variables.1045

OK, again, evaluating this function for an algebraic expression means I am going to substitute in 3b - 2 wherever I see an x.1054

Writing this out: this is a squared binomial...using the FOIL method, First is 3b times 3b, is 9b2.1074

The Outer terms--that gives me -6b; the Inner terms multiply to -6b; and the Last terms: -2 times -2 is 4.1093

Now, using the distributive property, and multiplying each term within the parentheses by 2,1106

is going to give me...2 times 9b2 is 18b2; 2 times -6b is -12b; and this is -12b;1111

I also could have just added these first, and then done the distributive property; it is going to come out the same either way.1124

2 times 4 is 8; now, the distributive property again here: -3 times 3b is going to give me -9b.1130

-3 times -2 is going to give me + 6; and then the 9.1143

Adding like terms: -12b and -12b is -24b; and I also have some more simplifying I can do out here.1148

This is 18b2; -24b and -9b is going to give me -33b; 8 and 6, plus 9 (that is 15 + 8) is 23.1160

OK, so evaluating this function for 3b - 2 gives this result.1177

We are simply substituting in 3b - 2 for all the x's, using the distributive property, and then simplifying by adding like terms.1184

OK, in Example 4, we are given this polynomial function and asked to describe the end behavior.1195

We are starting out by looking and making sure that this is written in descending order (which it is),1201

because I need to find the degree and the leading coefficient.1206

The degree is 6; I know that this is an even number, so it may or may not have zeroes, just as an aside.1210

Now, talking about the leading coefficient and the degree in a little more detail:1228

the degree being even tells me that both ends will be up, or both ends will be down.1243

OK, so I know that both ends are going to go in the same direction with this polynomial.1251

The leading coefficient here is 5, and that is positive; so since it is positive, both ends of the graph go upward,1256

just like with a parabola with a positive leading coefficient in the quadratic function.1278

This is to the sixth, so it is going to be more complex, and it may or may not have zeroes; but just schematically,1283

the idea is that both ends will face up.1295

Now, what does that tell me about the end behavior?1299

What this tells me about end behavior is that, when x is large, way out here, y is large (or the function is large).1301

When x is very small (way out here)--it has small values--y is large.1315

So, at both ends (where x is very large or x is very small), the function has large values.1323

And that tells me about the end behavior; and I was able to figure that out by realizing this is an even degree, and it has a positive leading coefficient.1330

That concludes this lesson of Educator.com on graphing and polynomials.1345