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INSTRUCTORS Carleen Eaton Grant Fraser Eric Smith
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Polynomial Inequalities

  • Remember that the solution to an inequality often involves a range of values.
  • To solve an inequality involving polynomials we
    • Set the inequality with zero on one side
    • Solve the related equation, with the polynomial equal to zero
    • Divide the x-axis into intervals using the solutions of the equation from step 2
    • Use test values from each interval to see if it satisfies the inequality
    • Check the endpoints of each interval to see if they need to be included
  • Do not attempt to split up the inequality over the factors. Even though it looks like we can use the principle of zero products we can’t. This is because we have an inequality and the principle of zero products only applies to equations.

Polynomial Inequalities

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
  • Objectives 0:07
  • Polynomial Inequalities 0:30
    • Solving Polynomial Inequalities
  • Example 1 2:45
  • Polynomial Inequalities Cont. 5:12
    • Larger Polynomials
    • Positive or Negative Intervals
  • Example 2 9:01
  • Example 3 13:53

Transcription: Polynomial Inequalities

Welcome back to 0000

In this lesson we are going to take a look at some polynomial inequalities. 0002

When it comes to polynomial inequalities, I’m going to show you two techniques that you can use to handle these.0008

One, you can take a look and figure out their graph and figure out where they are greater than or equal to 0 0015

or you can use a table to tracked down their sign.0020

Both of these are important and since we have developed lots of graphing techniques they are both very handy.0023

A polynomial inequality is simply an inequality involving a polynomial of some sort.0033

Down below I have many different examples of what I'm talking about.0040

Here is a nice -2x4 - x2 - 2 > 4 that will be an example of something we are trying to solve.0044

With this one, 8x3 + 2x2 < 6x + 5 is another good type of inequality that we could possibly find the solution to.0054

Depending on the type of polynomial present, sometimes you can call these linear inequalities, quadratic inequalities or even cubic inequalities.0064

I will not get too technical usually we will just call these like polynomial inequalities.0073

To solve one of these inequalities, it is best to get it in relation to 0. 0081

It means get everything over onto one side of your inequality symbol.0088

The reason why this is so important for the entire solving process is we are looking for those ranges of values that make the inequality true.0093

When it is all in relation to 0 then we just have to know where it is going to be positive or negative 0101

rather than trying to search for when it is greater, less than a some other number like 2 or 3.0106

That is much more difficult to do.0111

If you have a graph of the polynomial, this makes it especially easy because then you are essentially looking at whether it is above or below the x axis.0114

Let us take a linear equality like this.0128

We might know where it crosses the x axis everywhere where it is above the x axis we know it is positive.0132

Everywhere below the x axis we know the value is negative. 0141

If I had the equation for this and maybe I wanted to know where it was greater than 0 0147

then I can simply take all of the positive values where it is greater than the x axis.0157

Let us try a quick example to see this in action.0167

I want to use the graph of x2 - 3x -4 to see where it is greater than 0.0171

It is already said in relation to 0 for us, we do not have to work there. 0178

I just need a good rough sketch of what this looks like.0180

I can figure out where this thing is equal to 0 by using some of our factoring techniques.0186

Our first terms better be x and then - 4 + 1 will do fine.0193

I know it crosses the x axis at 4 and -1.0199

Since we have also done a lot with graphing quadratic like this one, we know which direction it is facing.0206

The (a) value is positive, so it is facing up.0212

Let us see what else do we know about this?0218

It looks like it has a y intercept of -4.0221

Let us put that in there.0226

A rough sketch of this might look something like that.0231

You will note that I did not go through all of the work to find the vertex because I’m not interested where it reaches its maximum or minimum value.0238

I'm interested in where is it above or below the x axis.0245

For this particular polynomial we want to know where it is greater than 0 because that is a way of saying where does it take on positive values.0258

We can see that it does this after the value of 4 and does it before the value of -1.0269

In those intervals, if you are using saying those x values, then the equation will be positive.0277

We will take those intervals as our solution, from negative infinity up to -1, and from 4 up to infinity.0289

That grabs both of those intervals.0301

We will use out little union symbol in between the show that it could be in either one of those.0304

That was a quadratic example but potentially you could use these for some much larger polynomials.0315

The key is creating an accurate enough graphs so we can see whether it is above or below the x axis.0321

You want it in relation to 0 to see what it is doing. 0328

I’m just going to give a rough sketch of a polynomial, so you can see how this process might work.0331

Let us call this my polynomial and I want to know where this one is less than or equal to 0.0341

Everywhere it is above the x axis would be positive and everywhere below the x axis it takes on negative values.0351

Look at these portions right here, here is below the x axis.0360

It has another little part right there and this last term.0366

We can highlight the intervals where you take on these negative values.0372

Here, here and here. 0377

Okay, now we just have to describe those intervals using our interval notation.0383

It is below the x axis from negative infinity all the way up to 1, 2, 3, 4, 5, -5 then it goes from -1 up to 3.0388

I got the 3, 4, 5 from 5 up to infinity.0405

We will use our little union symbol to connect all of those intervals. 0413

One quick note, we are including these end points because it takes on the value of 0 at these points.0417

We want them included.0427

Having a graph is handy. 0437

It gives us a nice visual way to see what is going on whether it is greater than 0 or less than 0,0439

but unfortunately we do not always have a graph to work with.0444

I’m thinking of some much more complicated polynomials.0447

In situations like this, we are going to use a table to keep track of the sign of its factors.0450

That way we will still be able to figure out whether it is above or below the x axis without ever seeing the graph.0455

Here is how this process works.0461

We will first get the inequality in relation to 0, so we will get everything over onto one side.0464

Then we will go ahead and solve the related equation. 0471

Instead of looking at the inequality we will throw in an equal sign in there and then solve it and figure out where it is equal to 0.0474

The reason why we are doing this is we want know where it could possibly change sign.0481

Where does it hit the x axis?0485

That will allow us to divide up the x axis into a lot of different intervals using those points. 0488

We will use our values around those points to see whether it is positive or negative in those factors. 0495

Once we have a lot of information about where is positive or negative 0504

we will determine which of those intervals actually satisfy the inequality.0507

We will check the end points of those intervals to see if we need to include them or not.0513

If we see something like or equals, we will usually include the values of our end points using brackets. 0520

If it is a nice strict inequality, then we will usually use parentheses to say that those end points are not included. 0532

Let us borrow a problem that we did earlier for quadratic and see how the same process works out now using a table.0542

The first thing we want is we want it in relation to 0, which fortunately it is already is.0552

We do not have to worry about that part.0558

I’m going to look at the corresponding equation.0561

Where could this thing equal 0?0563

I think we factored this one earlier it is going to come in handy once more.0568

I have x - 4 and x + 1.0572

There are two values this could equal 0, at 4 and -1.0578

Here is where our table comes into play. 0585

I’m going to make a number line and I’m going to mark out the location of -1 and the location of 4.0587

You can see that t does split up our number line into these different intervals. 0596

I need to figure out what is the polynomial doing on these different intervals.0602

I’m going to write down the factors of the polynomial and just experiment 0608

to see whether they are ending up being positive or negative on these intervals.0615

That is what our test points are for.0621

Let us give this a try.0623

I’m going to choose something on my number line that is less than -1.0625

Think of something like -2, something on that side.0630

We are going to test it into these factors over here.0634

If I was to plug in -2 in for x and then I subtract a 4, will that give me a positive value or negative value?0639

-2 -4 will be -6 and I know on this interval that factor is a negative. 0648

What if I took -2 and I put in the second factor, and I added 1 to it.0657

-2 + 1 I can see that that give me -1. 0662

Of course, the most important part is that factor would be negative on that interval.0667

I’m keeping track of the individual parts of the polynomial seeing whether positive or negative. 0672

Now that we have a good idea what is going on that interval, let us pick something on the next interval.0679

Something between -1 and 4.0684

Since I have worked out good, let us test out 0 and see what it does.0688

0 - 4 will that be positive or negative?0693

That will give us -4.0697

Let us plug in 0 into x + 1 and that will give us 1.0701

I know that it is positive on that interval.0706

Now something larger than 4, let us test out something over there.0711

Let us test out something like 5.0715

We will put that into each of our factors, 5 - 4 will give us 1 and 5 + 1 will give us 6.0718

We have lots of information about what each of these individual factors are doing on these intervals.0733

Of course, if we look back at the original problem that we are looking at, these factors are multiplied together.0739

Let us multiply these individual pieces together and least their signs and see what is happening to the overall polynomial.0746

If I were to take a negative × a negative, I would get a positive value. 0760

If I were to take a negative × a positive and I will get a negative value.0766

If I multiplied two positives together, I will get a positive value.0772

This is the sign of the overall polynomial when I put those pieces together.0777

What are my interests in, positive or negative values?0783

I'm looking for when the polynomial is greater than 0, so I want those positive values.0788

We will write down the value of the intervals where it was positive.0802

The things greater than 4 and things less than -1.0807

Note how I'm not including the end points here because we are dealing with a strict inequality.0821

This would represent the solution to our polynomial inequality.0827

Of course we might be doing with some more complicated polynomials and that is why we are using these techniques of a table.0834

I have 4x3 - 7x is less than or equal to 15x.0841

I have picked up a lot of the techniques for graphing this in a nice easy way other than just making a giant table of values. 0846

Let us use our table here, we would be able to figure out what the interval or range of solutions is for this one. 0853

I'm going to get everything over onto one side first. 0861

It looks like I'm looking where this is less than or equal to 0 or below that x axis.0872

We want to solve where this thing is equal to 0.0879

We will have to borrow some of our factoring techniques to help out.0886

Everything has an x in common.0891

Now we got all of those taken out. 0904

We can go ahead and factor the remaining quadratic.0908

What values could I use in here?0918

I have a 4x, 1x, 5, -3.0921

First terms will give us 4x2, outside would be -12, inside would be 5 and our last terms -15.0931

This checks out.0938

x = 0, 4x + 5= 0 and x – 3 = 0. 0941

We would end up solving each of these separately. 0950

This middle one, let us go ahead and move the 5 over and then divide by 4.0954

For the third one we will simply add 3 to both sides.0961

I have these 3 points where our polynomial could end up changing sign because that is where it is equal to 0.0968

Let us go ahead and draw our number line and put these values on there.0976

That way we can see how it divides up our number line.0980

The smallest thing we have here is -5/4.0985

The next largest would be 0.0990

The largest value we have is 3. 0993

Always put these from smallest to largest.0997

Let us also keep track of the pieces and our polynomials.1011

We have x, 4x + 5 and we have x – 3.1017

We are going to use our test value into these individual parts and put them together in the end.1025

Let us start off with our first test value.1032

We need something that is less than -5/4.1034

I know -5/4 is close to -1 I’m going to choose something a little bit farther.1039

It looks like I’m going to choose -2 and put it into all of our factors. 1046

If I put in -2 in 4x I will get -2.1050

If I put in -2 in for 4 then 4x + 5 that will give me -x + 5 which would still be negative.1056

Last that I could take that -2 and put it down into x -3, it will give me -5.1064

Negative again.1071

Let us choose something between -5/4 and 0. 1076

Here is where that -1 will come into play.1080

Putting a negative 1 in for x will give us a negative value. 1084

We need in 4 for 4x + 5 will give us 1.1089

Putting in for the last one, a -1 -3 = -4 so minus down there.1095

That takes care of that.1103

Let us see what happens when we put in 2 for all of these spots.1110

2 and 4x that will be positive.1115

Putting in 2 for 4x + 5 that will be 8 + 5 =13.1119

If I put in 2 for the last one, that will be 2 -3 = -1.1126


One more last thing, things that are greater than 3.1139

I’m going to test that out with 4.1143

Let us see if we can put that in all the spots.1147

I’m putting in for x and I will get 4 × 4 + 5 = 21 and 4 - 3 =1.1150

There would be that one.1160

That represents all of the individual components in the polynomial and what their sign will be on those intervals.1164

We want to imagine all of these being put together since all of these factors are being multiplied.1170

Let us do this one column at a time. 1178

Negative × negative × another negative, that value would be negative.1181

We have a negative × a positive and that will be negative, times another negative and that would be positive.1190

Now we have a positive × positive × negative, that is negative.1201

Our last column are all positives being multiplied together so positive.1207

I have a good sense of what the overall polynomial is doing for these different intervals.1213

I can see when it is negative, positive, negative and positive.1220

This problem right here, we want to know whether it is less than or equal to 0.1224

We are looking for the negative intervals.1231

You have this interval down here and we have this interval right here.1236

We can go back and highlight those on a number line.1241

I'm looking at the values all the way up to -5/4 and I’m looking at the values between 0 and 3.1246

Let us write down those intervals.1253

negative infinity up to -5/4 and from 0 to 3.1255

Note that I'm definitely including my end points here because this says or equal to 0.1266

I'm also interested in where my polynomial is equal to 0.1274

This would be our intervals solution.1280

No matter what method you use, it is often very important that you know where your polynomial is equal to 0.1284

Use many of our factoring techniques to help you out for that.1290

Then you can go ahead and look at the graph or even one of these tables to organize your information.1294

That will make your life much easier. 1298

Thank you for watching