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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

### Algebra 1 Online Course

### Transcription: Polynomial Inequalities

*Welcome back to www.educator.com.*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 -2x ^{4} - x^{2} - 2 > 4 that will be an example of something we are trying to solve.*0044

*With this one, 8x ^{3} + 2x^{2} < 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 x ^{2} - 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 4x ^{3} - 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 4x ^{2}, 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

*Negative.*1129

*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 www.educator.com.*1300

0 answers

Post by Sage Stark on August 12, 2017

Also, in example three, why did you change -5/4 to -2, it didn't here, but it could potentially alter the solution if the situation was right.

1 answer

Last reply by: Professor Eric Smith

Sun Dec 3, 2017 5:52 PM

Post by Sage Stark on August 12, 2017

In example three, why is the five positive and the three is negative? I thought it would be the other way around.