 Welcome to our review for the final exam for Math 1050 College Algebra for students at Southern Utah University. For one last time, I'll be your professor today, Dr. Andrew Misildine. And so now we've reached the end of the course. We're ready for the final exam for which it's there's some important information I want to say about this exam. There are many aspects of this exam that are going to be identical to previous exams we've taken in this class. For example, there's two types of questions you're going to see on this exam. You'll have multiple choice and free response questions. Those will be of the same format as we've seen before. As you prepare for this exam, you should prepare a three by five note card. Put whatever notes on it that you want. Even though the exam is going to be longer, your note card is not allowed to be bigger, still three inches by five inches. Feel free to write as small as you want. It just has to be handwritten and have pertinent mathematical information on it. You can write teeny tiny if you want. You can write on both sides. That's perfectly fine. You should prepare a note card as it's worth two points of that final exam. You can bring a calculator, scratch papers available. All that stuff is very, very similar. Some important differences, of course, with the final exam is the final exam is going to be longer. It's going to be 25 questions total. There are 15 multiple choice questions on this exam worth three points each, and then there will be 10 free response questions, which are worth five or six points based upon the difficulty of the question. This exam is a comprehensive exam covering every possible topic that we've tested upon throughout this course, plus a few things that showed up in the very last chapter that didn't appear on any of the previous exams. I'll talk about those things, and those new questions I'll point out specifically as we run across them. I want to utilize this video to go through the contents of this exam to help you better know what you should be studying for as you prepare for this final exam. In addition to the practice exam you see on the screen right now, I should point out that the past exams also provide to you a good study guide for this exam, because many of the questions on this final exam will be taken from previous exams, or at least variations of questions from previous exams. Your four practice exams, number one, two, and three are pertinent review guides for this final exam, plus also the actual exams one, two, three, four you took in this class as well. In addition to this practice exam you see on the screen right now, you have eight other exams you could use to help you study for this exam, and I would recommend you utilize all of those. So without further ado, let's get into the specifics of this exam content-wise. Things like time, place, and manner change from semester to semester, so please check with me to find out exactly where and when you're supposed to take this exam. All right, some questions are going to have a lot of variation to them, some will not. This very first question will be one of the type where there's not a lot of variation to this question. This first question will ask you to compute the average rate of change of a function. Well, the function might change and the interval will likely change as well. You will be asked to compute average rate of change. This question actually can be found on exam one, question eight, for which you were asked exactly something like this. I want to point out that as we're ending our semester with college algebra, college algebra, well, it serves many purposes. It is a general education mathematics class, but its primary purpose is as a prerequisite to more advanced mathematics classes. In particular, college algebra serves as a prerequisite to calculus. And so many times at other institutions, the class we're in right now might be called pre-calculus, again, as it's that's what its purpose really is meant to serve. And so a lot of the questions on this exam will be focused on getting us ready for calculus, because many of us will be taking that next semester or in the not too distant future, even if we don't realize it, we will be taking it. That might come upon us very, very soon. And so the notion of rates of change is very important calculus. And so this is a question we're going to see. Now on exam one, the questions here were quite mild, you know, mostly just like polynomials and the quadratics probably for the most part. I do want to warn you that for this final exam, all of the function families we've studied this semester are basically eligible on every question, even if it wasn't available on the first iteration of this question on a previous exam. So if you're asked to find the average rate of change, it could be a polynomial quadratic function like you see on the screen right now, but it could be a rational function, a radical function, an exponential function, a logarithmic function, a piecewise function. Any function we've studied this semester, this is all fair game. And that's kind of the whole point of our last unit in math 1050 here is we take all the things we've seen before, we mix them together now in a new cocktail that we haven't had before. And so be aware that you will be asked to compute the average rate of change, which remember that's just delta y over delta x on the interval a to b. This will just be f of b minus f of a all over b minus a. So your average rate of change is essentially just a slope formula where you take the y coordinate to be the function of the specific function, of course, will change and all of the function families we've seen will come up here. All right, moving on to question number two. Question number two will be a graphing type question. Again, very much focused on calculus in some degree, you'll be given the graph of a function f that you see illustrated here. You'll be asked something like is where's the function decreasing? Where is the function increasing? Where is the function concave up? Concave upward? Where is it concave downward? What is the domain of the function? What is the range of the function? Does the function have symmetry? Is it an even function? Is it an odd function? These are the types of questions are going to be asked again, given a graph represented geometrically, can we find these important questions from the graph? And so these things about like domain and the monotonicity, the concavity of the function, we asked similar questions this on exam one exam question three and 11 were questions we saw on exam one, the same type of thing will be given here where again, the function will be given as a graph. If we know a formula for it or not, doesn't really matter. I want you able to identify this information from the function graphically. Again, identical to questions we saw on exam number one. Question number three will be a question about logarithms. And in fact, you'll be asked to either expand the logarithms or you'll be asked to condense or combine the logarithms as well. This was a very important question we saw on exam four. It was actually question number four itself. There will be no variation on this except for what is the logarithmic expression you're given. Again, this is a version where we have to expand it. So you have a condensed logarithm, just a single logarithm. We want to expand it into something like this, but a variation of this question could be that you have the expanded logarithm and you want to condense it into something like this. Again, identical to question four from exam four as well. No variation on that one because that's a very important question. Again, as we're preparing for calculus. Question number four, you're going to be asked to, well, simply put, you're going to be asked to compute something. I know that's a very generic statement. I'll give you some more specific details in a second, but you're asked to compute something. Just a straightforward calculation of some kind. The version you see on the stream right now is you're asked to compute the quotient of two complex numbers, two minus two divided by two minus i. So this type of complex division, this was a question we saw on exam two, question three. We saw some other complex arithmetic like can we add complex numbers to track, multiply, divide, do some exponents with complex numbers? That was a question we saw on number 10. You could be asked to do that. You could be asked to compute a logarithm like what's log base two of eight or something like that. That was the first question from exam one. Excuse me, first question from exam four, be able to do that. Could we change the base? Could we compute a logarithm of the different base like we saw on question eight from exam number four? Could we evaluate a function? Can we add functions together? Can we subtract functions, divide functions, multiply functions? Those type of function computations, functions arithmetic. We saw questions like that on exam number one, question two, and question number four. So question number four should generally be considered a fairly easy question compared to other ones. You'll be asked to compute something like a complex number, logarithmic function, other simple function computations. Be prepared for something like that. Question number five is going to be an application, story problem of some kind. This will also be a set up the story problem. You don't necessarily have to solve it. What you see on the screen right now would be an example of a linear application, a linear story problem. This is what we saw for exam two, question eight, for which you have some quantity that's changing with a constant rate that leads to some type of linear function. An example of this that you don't see on the screen, that's a very important example to point out would be something like a simple interest or direct variation is another good example we could see right here. This could also be a exponential application. So we could do something like compounded interest. We could do, well, continuously compounded interest. Of course, we've had things like radioactive decay, exponential or population growth, Newton's law of cooling, logistic growth. All of those applications. So actually on exam four, question nine, we had a question identical to something that you could see right here as well. So we prepared for some type of story problem, probably a linear application or an exponential one, variation interest, those types of problems. One of those will be randomly selected for your version of the test. So you're going to want to prepare various set up the problems. Again, you don't need to solve it, just set it up. Question number six is going to be a question about function composition. That part is going to be guaranteed. So we wanted to do function composition, a very important aspect for calculus. Remember function composition, we have this little circle notation f of g of x. This means you have g inside of f. This function right here, they give you the composition and they give you one of the composition factors and you're supposed to find the other one. So this would be like function decomposition. But we saw on exam one questions like this, the particular one you see on the moment where you have to decompose it, that's like question number nine from exam number one. But question number seven was also apartment question where here's two functions, put them together, maybe evaluate them, something like that. Question number seven, it will involve absolute value. You might be asked to solve an absolute value equation like you see on the screen. That's similar to exam two, question two. But you also might be asked to solve a absolute value inequality. That was question number five from exam two, maybe graph an absolute value function. Question number seven will be a multiple choice about absolute value. Question number eight is going to be a multiple choice question about linear function in some degree, in some fashion. We had a lot of questions on exam number two that could do that. The version you see on the screen right now, it asks you to compute the slope of a line, be prepared to compute the slope. That was exam two, question number one. It's basically the same question as finding the average rate of change. So you should know that you should be able to solve a linear inequality. You should be able to solve a linear equation. You should be able to find the x and y intercepts of a line. You should be able to fit a line to two points like if I'm like, oh, what's the line that passes through these two points? Could you come up with the equation of that? These are all possibilities for question one from exam two, be prepared to do these basic calculations with linear functions. Question number nine is a very, very interesting question. There's not going to be any variation, but this is also a new question that we haven't seen on any previous exam. But it does relate to topics we've seen on previous exams. I should mention that. But the exact phrasing is a little bit new. And so what you're going to see here is you're going to be given a function f of x. And it's going to ask you two things, find all the x intercepts and find all the discontinuities of this function. The x intercepts are going to be things that solve the equation f of x equals zero, right? In particular, these things can nearly always be written as a fraction of some kind. Like notice you have an x and x, that's an x squared. This can be rewritten as one minus the natural log of x over x squared. When you have a fraction equal to zero, that happens if and only if the numerator one minus the natural log of x equals zero. So setting the numerator equal to zero and solving for that will give you the x intercepts. The discontinuities are going to come about by setting the denominator equal to zero. So if x squared equals zero, right, that might give you like a remove point or a vert glass until it or something. There's some type of break, a gap on the graph to some kind. We call it discontinuity because it's not continuous at that point. Finding the x intercepts and discontinuities of a function is an extremely important calculus question. And it's purely an algebraic thing, in which case you will then be asked to find these x intercepts and discontinuities. We'll list them together. So these are the x coordinates of those intercepts or discontinuities. So if you think it's this one, you'd select that choice. Now, don't worry about the functions themselves. I did select these functions f of x to be derivatives of functions you would you would find in a naturally in a calculus setting. So the these formulas might look a little interesting. But like I said, these are actually very natural calculus questions. The calculus part would be calculate the derivative, which I'm not going to talk about that. Because again, that's not part of our class. So this derivative is already calculated. But the next step is given the derivative find its x intercepts and discontinuities, or so to speak, finding the critical numbers of the original function. It's a very important question. You should be prepared to do that. And we talked about we had homework questions very much like this in our very last chapter. So please reference those for some more examples of what you could see on question number nine. For question number 10, we'll have some variability here. But this question will ask us something about a polynomial function. The current version you see on the screen is asking about the multiplicity of the roots, right? Which which are roots which touch the x axis but don't cross it. So that'd be like an even multiplicity. We could ask where are there odd multiplicities. We could ask what are the possible rational roots given a polynomial function. We could ask you to create a polynomial function given some information about roots and multiplicities and degrees using the fundamental theorem of algebra. We could solve polynomial inequality by factoring. We could expand something using the binomial theorem. There are a lot of questions from exam three very important here. So the question you see on the screen right now, this is actually question one from exam three about multiplicities. You could be asked about the possible rational roots. I said that's question four from exam three. You could be asked about the fundamental theorem of algebra. That was question number six. The binomial theorem was question number seven. Solve a polynomial inequality. Like I said, question number nine from exam three. So question ten will be about polynomials and will be some version of a question from exam three. Please reference them for some more practice. In that same vein, question number 11 will be a question about rational functions also coming from exam number three. So the version you see on the screen right here is you're given a rational function asked to find its vertical asymptote. You could be asked to find what's one of the horizontal asymptotes of this rational function. What's the domain of the rational function? Does it have a remove point? Is there an oblique asymptote? Just again, I'd solve a rational inequality just to give you some examples. So again, the version you see on the screen right now, this is question eight from exam three about finding vertical asymptotes of domains of a rational function. Those things are related. We could find a horizontal asymptote that was question number two from exam three. An oblique asymptote was question number five from exam three. Solving rational inequalities was question nine from exam three. So again, you'll get some question from exam three about rational functions. Okay. Then the next one, question number 12 is going to be something about exponents, some type of power question. It could be a question about solving or working with a power function, but on the flip side, it could also ask about exponential functions. It could have to do with logarithms as well because exponentials and logarithms and power functions all have to do with how exponents work. So to solve a question like number 12, you probably want to use things like exponential laws, logarithmic laws to help you simplify these things. The version you see on the screen right now is solving an exponential equation where you have a common base. Since exponential functions are one to one, you can utilize a property to help you out there. So this is a version of question number two from exam four. You might have to answer, you might have to solve an equation, a simple logarithmic equation like question six from exam four. You might have to fit some points inside of an exponential curve that has build an exponential function. We did that on question number three from exam four. It could also be a logarithmic function. You could be asked to graph maybe a logarithmic function or an exponential function solving. Another question from exam four is question number seven that had to do with solving logarithmic and exponential equations. Also statements about power functions we saw from exam three. These are all possibilities you could see on question number 12. Question number 13 moving on to the next page will be another question that will be about graphs. The way that these exam questions are typeset, graphs generally take up a lot of space on the screen and therefore I've put a lot of the randomization so that the graph questions are together so that when the test is randomized it doesn't have weird formatting issues. So question number 13 will be definitely a question about graphing of some kind. So you could have a question like the very first question on exam one about is this graph a graph of a function? Does it pass the vertical line test? You could have a question like number five from exam one about is this graph one to one? Does it pass the horizontal line test? We had a question on exam one about piecewise functions. The question you see on the screen right now would be similar to question number 10, although this is a graphical variant than the algebraic one we saw on exam number one about symmetry. Does this graph demonstrate any type of symmetry? Is it symmetric with the x-axis, y-axis? The origin is an even or odd function. Same type of thing there. Another type of question that you're going to see that we haven't seen on any previous exam is just about function identification. Like if I give you a graph of a function could you tell me like oh that's the graph of x cubed or that's the graph of the square root of x or that's the graph of e to the x or that's the graph of the log base 2 of x. There are certain very specific functions that we have been expected to memorize. So there are some power functions that we should know like x to the n. So this will be things like x squared, x cubed, x to the fourth. Do we know these monomial functions? What about the reciprocal functions 1 over x, 1 over x squared, 1 over x cubed. All of these can be written as x to the negative n. You also have the radical functions like the square root of x, the cube root of x, the fourth root of x. These of course could all be written as x to the 1 over n power. These are all examples of power functions and we should know each and every one of these graphs. It's important to throw in the absolute value function. That's a very special piecewise function that you should know the graph by memory. We should also talk about exponentials and logarithms. Can we graph these functions? Just by the information that's here on the screen, these are important function families and you should know them essentially by memory. Like if I gave you the graph of x to the fifth, could you distinguish that it's an odd monomial function as opposed to some others? So be able to correctly identify a function from this library of functions is also a variant that we haven't seen on a past exam but you could see for the final exam. That's a very good comprehensive question. Be prepared for something like that. We draw close to the end of the multiple choice section. Question number 14 will be a question about a quadratic function of some kind. So you might be given a quadratic function and asked to find the vertex of the quadratic function. Maybe you're asked to convert it to the vertex form. You could be asked to solve a quadratic equation using maybe the quadratic formula completing the square by factoring. These are all versions of question four from exam two that we've seen previously. You might be asked about the discriminant of a quadratic function. This is like question six from exam two. If your discriminant is positive, you'll have two real solutions to the equation. If it's discriminant zero, there's one real solution. If the discriminant is negative, there's no real solutions, those type of things. Could you solve a quadratic inequality? This is like we did with question nine from exam two. So question number 14 will ask you to do something with the quadratic functions. Question number 15, our last question on the multiple choice section of the final exam will have to do with systems of linear equations. Essentially, this will be question seven from exam two for which we might be asked something about is this a solution to the system? How would you solve this using elimination or substitution? If you used an augmented matrix, what does this tell you? Is it consistent? Are there dependent and multiple solutions? Some question about linear systems you will see on question number 15. So before we start the free response, let me just kind of summarize what we've seen here that while there's a lot of randomness to the questions, we saw this in the multiple choice section, the randomness is designed in such a way that specific topics will come up. Like question 15, you will see a question about linear systems. Question 14, you will see a question about quadratics. Previous questions, you will see graphing questions. You will see algebraic questions. You will have questions about exponentials and logs. You will have about rationals and polynomials and linear functions. So while the specific question type might change here or there, you can't avoid really any of the topics as you're preparing for it because you will have to do something about linear functions, polynomials, etc. All right. So with that, I want to then start the free response section of this exam. There's 10 more questions to discuss. Question number 16 will be about polynomial division. It could be long division of polynomials like you see on the screen right now. It could be about synthetic division. We had two questions on exam, three about these things. Question 10 was like the one you see on the screen. Question 13 was about synthetic division. That's a variation you could see. This question will be worth five points. Question number 17 will be a story problem of some kind, but unlike the one we saw in the multiple choice section, this one we have to set up and we have to solve it. The version we see on the screen right now is actually question 11 from exam two, where we have to set up and solve a system of linear equations for a mixture problem, right? Mrs. Jones has two different investments at different interest rates. So set up the system and solve it. So you might have to do something like that. We might also have a question like quadratic applications we saw with question number 14 from exam two. So things about projectile motion, maximizing area. Essentially, you set up a quadratic function and use the vertex to find the maximum. Things about revenue, right? Revenue is equal to the number of things you have times their price. Can you maximize revenue? Well, price is a linear relationship on x, which often it is. This gives you a quadratic function. You find the vertex that gives you the maximum revenue. We also could have various polynomial or rational applications like we saw, we saw these at the very end of chapter five, where we had things about finding volumes of boxes that often led to a polynomial application. Various rates of change naturally lead themselves to rational applications, things like inverse variation, joint variation. These lead to polynomial and rational applications. Be prepared for something like that. We didn't really exactly see very many of these on exam three, but we did have homework on them. And the final exam is potentially going to ask us something about that. It won't be too crazy. It won't be worse. It wouldn't be worse than, of course, any type of quadratic, any quadratic, any application, excuse me. In particular, it's one that you can solve without a graphing calculator. So don't worry about some of those harder ones we saw on that homework assignment. Question number 18 is going to ask us to solve an equation. The type of equation will be basically anything. The version you see on the screen right now is a radical equation like we saw on exam three, question 11. Can you solve an equation involving square roots, cube roots, et cetera? But we might have to solve equations involving quadratic equations like question 12 from exam number two. We might have to solve rational equations from exam three, exponential or logarithmic equations like exam four, 11, and four, 13. But there's also a new type of question that we might see on this exam where what if we have some type of quadratic like equation? So what if we had something like the natural log of x squared, let's say minus the natural log of x plus, let's say, no, we'll do negative, negative six equals zero, something like that. In this situation, we have a quadratic like equation. Notice if we say, oh, the natural log of x, if I set that to equal to u, then this equation becomes u squared minus u minus six, like so, for which then we can factor that as u minus three and u plus two. Right. Notice negative three times two is negative six, negative three plus two is negative one, that's the correct factorization. So we end up with u equals three or negative two. Removing the variable u, you end up with the natural log of x is equal to three or negative two, hence x is equal to e cubed or e to the negative two. And therefore we can solve this so-called quadratic like equation using quadratic methods, but then combining it with other functions like natural log or honestly whatever. We should be prepared how to solve these quadratic like equations, which we did see in chapter seven on pre-calculus. Moving on to question number 19, which is a two point, sorry, six point question here. You'll be asked to set up and simplify a difference quotient, something of the form f of x plus h minus f of x equals h. We had a question like this back on exam one question 14, I think it was. I might not know the exact number off the top of my head, sorry about that, but there was a question on exam one about setting up and simplifying difference quotients. It mostly had to do with quadratic difference quotients, but be aware that as we've gone through more and more functions, there's more and more difference quotients that we can simplify. In fact, our very last lecture is about, lecture 50 is about difference quotients and the homework attached to lecture 50, aka homework 50, gives us lots of examples of the types of difference quotients we should be able to set up and simplify. So for example, what if our function was given as something like f of x equals the square root of x, then our difference quotient f of x plus h minus f of x all over h would then look like the square root of x plus h minus the square root of x all over h. We have to also be able to set up and simplify things like this. For a square root, of course, we're going to times the top and bottom by its conjugate at the square root of x plus h plus the square root of x, we're going to times the bottom by that same value, the square root of x plus h plus the square root of x. And then we work from there trying to simplify. Again, this is just one of many examples. It's very important that we be able to set up and simplify these difference quotients. This is probably one of the most important problems we should be studying as we move forward towards calculus, because this is honestly one of the first problems one deals with with calculus, setting up and simplifying these difference quotients. So if there's any question you study for for this exam, make sure you cover question 19. Question number 20 will be a question asking us to graph a function of some kind. So it could be a polynomial function like we saw on question 14 from exam three, it could be a rational function like we see right here. This would be like question 15 from exam three. So that we're going to graph functions using multiplicities of x intercepts graph the y intercept graph horizontal asymptotes or oblique asymptotes or whatever the in behavior is maybe using the leading term if it's a polynomial graphing vertical asymptotes identified their multiplicities and you're going to put all of that over here. We did this back in chapter three, but in our final chapter, we're also introducing this possibility with what if our multiplicity is a fraction? What if we have something like x to the one half or x minus one to the one third power? What does this do to the shape of the graph? You still have an even and odd multiplicity, right? So this one's going to touch, this one's going to cross that business is the same. But now that our multiplicity is less than one, we might have unusual steepness, right? Like if you were graphing just the function y equals x minus one to the one third, you move the graph to the right so that the x intercept becomes one, you see it right here. But then at the x intercept, you see this unusual steepness. It's really steep right here at the x intercept as opposed to other polynomial functions. It's much flatter at the x intercept, right? Or maybe get something like that. It's much flatter closer to a line. This one, it's like basically up and down when you cross the x intercept. That's what radicals are going to do. And it crosses the x, it crosses the x axis at that location. Okay, so you should be prepared to graph something with that type of behavior, maybe a cusp or something like that. These are some examples of algebraic functions we saw in our very last chapter on pre-calculus. Question 21, you'll be asked to compute the inverse of a function algebraically. This is identical to question 13 from exam one. You asked, you were asked to do this exact thing. We've revisited this very question on 12 from exam number three, where now the family of functions got expanded. And on exam one, we basically only dealt with very simple algebraic functions comparatively, maybe like polynomials, rationals, but in exam three, we didn't allow for exponents and radicals to come into place. On the final exam, you might also involve exponentials and logarithms, as those are functions which are one to one and thus have inverses. We could change this very question before e to the x minus two over three e to the x plus one. That little change does not fundamentally change how you calculate the inverse. You should be prepared to answer questions of that form. Question 22, you'll be asked to find the domain of a function. And again, everything goes right here. When finding the domain of a function, the things we should watch out for is if you take the square root of a negative, that gives you something imaginary. So any even radicals, we have to make sure the radicans are not negative. If you have any division by zero, that will give you a vertical acetote or best case scenario remove point. That's a problem with the domain. That's something probably going on with this function right here. Be cautious of that. If we take the log of zero or the log of a negative, this actually turns into the same problems that we see right here. Any log of zero is equivalent to a vertical acetote. It's like dividing by zero and taking the log of a negative is much like taking the square root of a negative. It leads to some imaginary numbers. So could you find the domain of a function? This could be a algebraic function, a transcendental function, or a combination of these. Any function we possibly could have seen this semester goes for question 22 here. Now on exam one, you had a question similar to this, but things were very much tame. Later on in exam four, question five, actually, we have one where you have to find the domain of a logarithm. Be aware that finding domains often leads to solving inequalities, because if you have a square root or you have a natural log or any log, honestly, then we need that the argument of the square root is greater than equal to zero. And for the logones, we need that its argument is greater than zero. So oftentimes finding domain comes down to solving inequalities. And those inequalities could be quadratic inequalities, linear inequalities, absolute value inequalities. You name it, it could be there. So be prepared for a lot of possibilities here. Question 23, you'll be asked to graph a function using transformations. We had a question like this on exam number one, question 16, in fact. On question 16, you were given the graph. So the original graphs, it's like, oh, transform y equals x squared using some transformations, right? It gave you the initial graph. That's not going to be there this time, because you were expected to know the graphs of x squared, x cubed, the square root of x, one over x cubed, e to the x, the natural log, those type of things. And yes, logarithms, exponentials are viable options for this question. I should mention that exam two also had a question similar to this question 15. That one was specifically on quadratic functions. It had the element that the function was not in the transformation form. You had to complete the square to put in the vertex form for which then the transformations became readily available. We also had questions on exam four similar to this, like question 410. You had to graph a exponential or logarithmic function. You used transformations to do that. Just like on exam one, you must indicate the transformations. Did you shift upwards or downwards, right, left? Did you stretch it vertically, horizontally? Did you reflect anything? All of those things should be mentioned. And graph at least one, two, three points on the graph. So it becomes very clear what it is the graph you're plotting. Question 24 will ask you to solve a system of equations to some degree or another. You might be asked to solve a three by three system of equations, like we saw on exam two, question number three. You might be asked to compute the partial fraction decomposition of some rational function. This was the very last question of exam four, question 15. We often avoided the systems of linear equations because I typically use a technique of annihilation, but all of those partial fraction decompositions naturally lead to systems of linear equations. And for that reason, I'm coupling it with this one. But you also might see a new type of problem that we didn't see on any of the midterm exams, specifically what you see on the screen right now. You might have a system of nonlinear equations for which you'll only have two equations and two unknowns, but there could be some type of quadratic relationship going on with X or Y. So like in this situation, we have a line intersecting a parabola that could lead to multiple solutions. And we saw these, of course, on homework 46. So you might want to practice some of those. This will be worth six points. And then the very last question, question 25, you'll be given a high degree polynomial. So this could be like degree three, degree four, degree five, degree six, probably not bigger than degree six. That's pretty big, honestly, for this one. In which case you need to find all of the real and complex roots, sometimes called zeros of the polynomial. This is this advanced factoring problem. So you'll use things like the rational roots theorem, Descartes rule of variation of signs, upper and lower bound theorems, just to name a few of the tools in the toolbox that you could use here. The idea is we want to factor this and find all of the roots with their multiplicities. And if there are any, if there's any non real roots, you need to list those as well. So this question 25, the last question on your exam will be identical to the very last question on exam three, question number 16. No variation on this one. This is a very good example, very good question to test really, are your algebraic skills mature enough to pass this class and move on? Or do you need more time to develop them? There's a lot of stuff that goes on. And that's the thing is a student who can do well on a question like number 25 really indicates that they've become a mature algebraic student and is again ready to move on. This question's worth six points because again, it's very, very involved compared to maybe some of the other ones. And so that brings us to the end of this exam. Like I said, it's going to be longer, but you'll have more time to take it of course, be patient, do the best you can, study, let me know if you have any questions. But I can tell you that I know you're going to do great. If you've made it to this point of the semester that you're actually sitting here and watching the end of this 40 minute video, it means you're in a really good place in this class. I imagine that you're going to do great on this exam, believe in yourself, and that's going to be the key. Confidence is the most important algebraic skill out there. If you believe you can do mathematics, then you can do mathematics. And I believe in you, start believing yourself if you haven't already done so. So congratulations to making it to the end of the semester. It's been pretty fun. I hope you the best on this exam, and maybe I'll see you next time in calculus or some other mathematics class. Anyways, have a good day. Best of luck. Bye.