“basic units” of algebra [Allen]. These concepts undoubtedly, integrated and related many areas of mathematics, amongst these, topology, theory and analysis [Allen]. The Fundamental Theorem of Algebra As the study of algebra became more expansive and necessary, it became of extreme importance to many other disciplines as well. And as remarkable mathematicians made new discoveries, algebra developed and progressed as a mathematical science. Prodigious individuals such as Descartes and Fermat were founders in the creation and development of analytical geometry, including theory on polynomials. Polynomial theory, thanks to Descartes, then became a topic of great interest. Therefore, algebra came to be identified with the “theory of polynomials” [Corry]. And so, the need to solve equations of a higher order than four, gave rise to the Fundamental Theorem of Algebra. The theorem states: “Every polynomial equation of degree n with complex coefficients has n roots in the complex numbers” [FTA]. Many mathematicians, such as, Leibniz, Euler, Laplace and D’Alembert attempted to prove this to no avail. It was not until 1799 that Gauss, Fig 1, provided the first complete proof to this theorem in his doctoral dissertation [FTA]. Gausses’ efforts led to the complex numbers of the form a + bi, which in turn led to fields and later to rings [Allen]. Figure 1, Carl Friedrich Gauss [4] Conclusion Algebra, as it is known today, encompasses a wide range of mathematics. In essence, it…
1. What are the least, and most, amount of distinct zeroes of a 7th degree polynomial, given that at least one root is a complex number? Answer: If the equation is 7th degree then it has 7 roots. Those roots can be complex or real. Complex roots always come in pairs, so if it has one, then it has 2, the other one being the conjugate of the first one. This in other words, if one complex root is a + bi, then the other complex root is a – bi. If at least one root were complex, then we would have a…
In 1540, a man by the name of Lodovico Ferrari, please be aware that I don’t think his name has anything to do with the sports car, was an Italian mathematician known for discovering the solutions to quartic functions. A quartic function is a function of the form ax^4 + bx^3 +cx^2 +dx+e, where a is a nonzero, which is defined by a polynomial raised to the fourth degree, called quartic polynomial. We will probably go more in depth about these quartic polynomials soon in class. My quartic…
Buried Treasure Ashford University MAT 221 Buried Treasure For this week’s Assignment we are given a word problem involving buried treasure and the use of the Pythagorean Theorem. We will use many different ways to attempt to factor down the three quadratic expressions which is in this problem. The problem is as, ““Ahmed has half of a treasure map, which indicates that the treasure is buried in the desert 2x + 6 paces from Castle Rock. Vanessa has the other half of…
Entering the labor market or continuing education beyond a certain point is a very important individual level investment decision. An important determinant of the demand for education is its expected benefits. The benefits depend upon the value of an individual’s labour input, which in turn depends upon the level of education. Hence, the education-wage relationship can be used to measure the returns to schooling. The rural and urban sectors differ widely in terms of the education and employment…
Figure 4, shows the oscillation of our drop test. Once converting the millivolts to position we were able to develop a new polynomial curve of position vs. time of the free fall motion of the block, this is illustrated in Figure 5. To calculate the acceleration we must first the velocity of points in our data over a given period of time. To do this we used the equation, v= ∆x/∆t …
The Mandelbrot and Filled Julia Sets The Mandelbrot and Filled Julia Sets are sets of numbers that are composed of complex numbers that do not diverge for varying conditions of the function fc(z)=z²+c. For the Mandelbrot set, the function is iterated from z=0. The filled Julia set holds c constant for any complex number to look at the behavior of the iterated functions for each value of z in the complex plane. Both sets are derived from the iterations of simple functions in the complex plane,…
Semra Özal CONNECTIONS OF LOGARITHMIC FUNCTIONS Logarithm and exponential functions have close relationship and they are inverse function of each other in a way. Before explicitly clarifying this inverse relationship, we should analyze their definitions. Logarithm means, in mathematics, “The exponent that indicates the power to which a base number is raised to produce a given number “2 Exponential function means that “mathematical function in which an…
In this lab we will explain how to deduce the equation landing position of the ball thrown by the robotic arm stand. We must create the equations in terms of the radius, height, arm rotation speed, and release angle. The equations must be made with those variables because, those variables are what can change where the ball will land in the cup, and will make the equation more useful to the client. To make sure the equation will remain accurate we have to keep track of all the units throughout…
Algebra was never my favorite course to hear about in school. With algebra being a necessity for my degree program it’s going to be challenging. Though it will be challenging, I know it is very much needed to cause me to be successful. I’m willing to push through the pressure to get to the prize. Algebra has always challenged me, even in school. The fact that being forced to take algebra is a challenge, I’m well-prepared for the ride. No matter where I go, I understand that algebra is needed to…