![]() ![]() If you want to calculate the definite integral from 0 to infinity (which I guess the original responder was referring to), it is very useful to consider complex analysis, and the response starting with y = x 5 + 1 is just not the way to go. It's a fun geometry/trig exercise to find it using a clever isosceles triangle or some trig double/triple angle formulas.ĭude, really? Besides that what you are writing is just plain wrong, the original response was about complex analysis. This lets you write everything down more "numerically", which is why the solution Wolfram shows radicals inside the ln and arctan functions instead of sin and cos of multiples of 2pi/5. Incidentally, there is also a geometry trick to rewrite the coordinates as radicals - sort of like the "exact values" tricks you learn when finding the sin and cos of pi/6, pi/4, pi/3, etc. Having all the complex roots lets you factor the nasty quartic over R - just multiply the factors (x-z)(x-z*) corresponding to complex conjugate solutions to get the two irreducible quadratics that the quartic factors into. Alternatively, it's a well-known fact from field theory that the nth roots of -1 sit at the vertices of a regular n-gon in the complex plane, so you can use trig to find their coordinates. There's the obvious root x=-1, and so you can use long division to factor out x+1, but then you get a nasty quartic with no obvious factorization strategy. You don't have to, but I don't know of an obvious way of factoring x^5+1 over R. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License. ![]() Use the information below to generate a citation. Then you must include on every digital page view the following attribution: If you are redistributing all or part of this book in a digital format, Then you must include on every physical page the following attribution: If you are redistributing all or part of this book in a print format, Want to cite, share, or modify this book? This book uses the In this chapter, we will explore these kinds of relationships and their properties. ![]() For any year we choose, we can determine the corresponding value of the stock market average. Notice, as we consider this example, that there is a definite relationship between the year and stock market average. The result caused the sharp decline represented on the graph beginning at the end of 2000. Many companies grew too fast and then suddenly went out of business. As bubbles tend to do, though, the dot-com bubble eventually burst. That five-year period became known as the “dot-com bubble” because so many internet startups were formed. It shows that an investment that was worth less than $500 until about 1995 skyrocketed up to about $1,100 by the beginning of 2000. tracks the value of that initial investment of just under $100 over the 40 years. As a result, the Standard and Poor’s stock market average rose as well. Toward the end of the twentieth century, the values of stocks of internet and technology companies rose dramatically. ![]()
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