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The Complete Theory of Equations: Powerful Concepts and Techniques (20+ Solved Examples)

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The Theory of Equations is a cornerstone of mathematics, offering tools to find and understand the roots of polynomial equations and their relationship to coefficients. This blog explores key concepts such as symmetrical functions of roots, surd and imaginary roots, De-Cardes’ rule of signs, and Horner’s process for polynomial evaluation. Dive into advanced topics like transforming equations, odd and even degree properties, and solving cubic and quartic equations with methods like Ferrari’s. Whether solving algebraic problems or understanding polynomial behavior, this guide equips you with essential insights and techniques.

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Legendre Polynomial

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Legendre Polynomial Show that the coefficient of ( x^n ) in ( P_n(x) ) is ( frac{(2n)!}{2^n(n!)^2} ) Solution: The n-th Legendre Polynomial is given by [ P_n(x)=frac{1}{2^nn!}frac{d^n}{dx^n}left( x^2-1right)^n ] Expanding ( left( x^2-1 right) ) using Binomial theorem, we get begin{align*} P_n(x) &=frac{1}{2^nn!}frac{d^n}{dx^n}left[ x^{2n}-nx^{2(n-1)}+cdots right]\[0.2cm] P_n(x) &=frac{1}{2^nn!}left[ frac{d^n}{dx^n} left( x^{2n}right) -cdots right]\[0.2cm] P_n(x) &=frac{1}{2^nn!}left[

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Taylor Series and Maclaurin Series Explained: Derivations of Maclaurin Series of 10 Essential Functions

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Taylor Series and Maclaurin Series Explained: Derivations of Maclaurin Series of 10 Essential Functions Table of Contents   Introduction Calculus, the language of change, equips us with powerful tools to analyze and understand how things move, grow, and interact. Imagine a scenario where you need to determine the future position of a speeding car or

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LaTeX Unlocked: A Powerful Blueprint for Beginners to Conquer Document Design! (Part 2)

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LaTeX Unlocked: A Powerful Blueprint for Beginners to Conquer Document Design! (Part 2) Table of Contents   Introduction In the first part of this blog, we covered the fundamentals of LaTeX, from installation and basic structure to document formatting and typography. Building on this foundation, the second section of our investigation delves into more advanced

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LaTeX Unlocked: A Powerful Blueprint for Beginners to Conquer Document Design! (Part 1)

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LaTeX Unlocked: A Powerful Blueprint for Beginners to Conquer Document Design! (Part 1) Your Web Page Introduction Welcome to the world of LaTeX, a sophisticated typesetting system that turns your writings into polished, professional works of art. LaTeX provides a versatile and fast way to write documents with precision and flair, whether you’re a student,

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From Fundamentals to Advanced: 50+ Integration Formulas Explored

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From Fundamentals to Advanced: 50+ Integration Formulas Explored Your Web Page Table of Contents Introduction Unlock the power of mathematical analysis with Integration Formulas, a crucial element in calculus. Integration is a mathematical operation that involves finding the integral of a function, representing the area under its curve on a given interval. It is the

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