Algebra
Archit Pal Singh Sachdeva
1. Consider the sequence of polynomials deﬁned by P1 (x) = x2 − 2 and Pj (x) = P1 (Pj−1 (x)) for j = 2, 3, . . .. Show that for any positive integer n the roots of equation Pn (x) = x are all real and distinct. 2. Prove that every polynomial over integers has a nonzero polynomial multiple whose exponents are all divisible by 2012. 3. Let fn (x) denote the Fibonacci polynomial, which is deﬁned by f1 = 1, f2 = x, fn = xfn−1 + fn−2 . Prove that the inequality 2 fn ≤ (x2 + 1)2 (x2 + 2)n−3

holds for every real x and n ≥ 3. 4. Find all polynomials f with with real coeﬃcients satisfying, for any real number x, the relation f (x)f (2x2 ) = f (2x3 + x). 5. Consider the equation with real coeﬃcients x6 + ax5 + bx4 + cx3 + bx2 + ax + 1 = 0, and denote by x1 , x2 , . . . , x6 the roots of the equation. Prove that 6

k=1

(x2 + 1) = (2a − c)2 . k

6. Let a, b, c, d be real numbers such that (a2 + 1)(b2 + 1)(c2 + 1)(d2 + 1) = 16. Prove that −3 ≤ ab + bc + cd + da + ac + bd − abcd ≤ 5. 7. Solve the equation x3 − 3x = 1 √ x + 2.

8. For positive numbers a, b, c prove the inequality a2 − ab + b2 + b2 − bc + c2 ≥ a2 + ac + c2 .

9. Let a, b, c be real numbers. Show that a ≥ 0, b ≥ 0, and c ≥ 0 if and only if a + b + c ≥ 0, ab + bc + ca ≥ 0, and abc ≥ 0. 10. Find the minimum possible value of a2 + b2 if a and b are real numbers such that x4 + ax3 + bx2 + ax + 1 = 0 has at least one real root. 11. If x and y are positive real numbers such that (x + x2 + 1)(y + y 2 + 1) = 2011, ﬁnd the minimum possible value of x + y. 12. Let p ≥ 5 be a prime number. Prove that 43 divides 7p − 6p − 1.

...Algebra is a way of working with numbers and signs to answer a mathematical problem (a question using numbers)
As a single word, "algebra" can mean[1]:
* Use of letters and symbols to represent values and their relations, especially for solving equations. This is also called "Elementary algebra". Historically, this was the meaning in pure mathematics too, like seen in "fundamental theorem of algebra", but not now.
* In modern pure mathematics,
* a major branch of mathematics which studies relations and operations. It's sometimes called abstract algebra, or "modern algebra" to distinguish it from elementary algebra.
* a mathematical structure as a "linear" ring, is also called "algebra," or sometimes "algebra over a field", to distinguish it from its generalizations.
A variable is a letter or symbol that takes place of a number in Algebra. Common symbols used are a, x, y, θ, and λ. The letters x and y are commonly used, but remember that any other symbols would work just as well.
Variables are used in algebra as placeholders for unknown numbers. If you see "3 + x", don't panic! All this means is that we are adding a number who's value we don't yet know.
Term: A term is a number or a variable or the product of a number and a variable(s).
An expression is two or more terms, with operations...

...HISTORY OF ALGEBRAAlgebra may divided into "classical algebra" (equation solving or "find the unknown number" problems) and "abstract algebra", also called "modern algebra" (the study of groups, rings, and fields). Classical algebra has been developed over a period of 4000 years. Abstract algebra has only appeared in the last 200 years.
The development of algebra is outlined in these notes under the following headings: Egyptian algebra, Babylonian algebra, Greek geometric algebra, Diophantine algebra, Hindu algebra, Arabic algebra, European algebra since 1500, and modern algebra. Since algebra grows out of arithmetic, recognition of new numbers - irrationals, zero, negative numbers, and complex numbers - is an important part of its history.
The development of algebraic notation progressed through three stages: the rhetorical (or verbal) stage, the syncopated stage (in which abbreviated words were used), and the symbolic stage with which we are all familiar.
The materials presented here are adapted from many sources including Burton, Kline's Mathematical Development From Ancient to Modern Times, Boyer's A History of Mathematics , and the essay on "The History of Algebra" by Baumgart in Historical Topics for the...

... alone.
= 5 The answer after simplifying.
+ -
3 - x – 5 The final answer.
In conclusion, I have found polynomials are very beneficial to one’s daily life. The use of algebraic functions is very common without individuals actually knowing it. It makes handling business matters easier. When an individual wants to learn about investing or saving, polynomials will be an easy way to receive the exact amount time and what you will need to reach your goals. Until this class, I would have never known how important algebra can be to one’s daily life. Once I began the logics it began to make sense. Now I plan to use this formula normally from this day forward.
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...Unlike geometry, algebra was not developed in Europe. Algebra was actually discovered (or developed) in the Arab countries along side geometry. Many mathematicians worked and developed the system of math to be known as the algebra of today. European countries did not obtain information on algebra until relatively later years of the 12th century. After algebra was discovered in Europe, mathematicians put the information to use in very remarkable ways. Also, algebraic and geometric ways of thinking were considered to be two separate parts of math and were not unified until the mid 17th century.
The simplest forms of equations in algebra were actually discovered 2,200 years before Mohamed was born. Ahmes wrote the Rhind Papyrus that described the Egyptian mathematic system of division and multiplication. Pythagoras, Euclid, Archimedes, Erasasth, and other great mathematicians followed Ahmes ("Letters"). Although not very important to the development of algebra, Archimedes (212BC 281BC), a Greek mathematician, worked on calculus equations and used geometric proofs to prove the theories of mathematics ("Archimedes").
Although little is known about him, Diophantus (200AD 284AD), an ancient Greek mathematician, studied equations with variables, starting the equations of algebra that we know today. Diophantus is often known as the "father of...

...Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis.
For historical reasons, the word "algebra" has several related meanings in mathematics, as a single word or with qualifiers.
• As a single word without article, "algebra" names a broad part of mathematics (see below).
• As a single word with article or in plural, "algebra" denotes a specific mathematical structure. Seealgebra (ring theory) and algebra over a field.
• With a qualifier, there is the same distinction:
• Without article, it means a part of algebra, like linear algebra, elementary algebra (the symbol-manipulation rules taught in elementary courses of mathematics as part of primary and secondary education), or abstract algebra (the study of the algebraic structures for themselves).
• With an article, it means an instance of some abstract structure, like a Lie algebra or an associative algebra.
• Frequently both meanings exist for the same qualifier, like in the sentence: Commutative algebra is the study of commutative rings, that all arecommutative algebras over the integers.
• Sometimes "algebra" is also used to denote the operations and methods related to algebra in the study of a structure that does not belong to...

...ALGEBRA
In all three of these problems there is use of all of the terms required: simplify, like terms, coefficient, distribution, and removing parentheses. There is also use with the real number properties of the commutative property of addition and the commutative property of multiplication. In what ways are the properties of real numbers useful for simplifying algebraic expression? The properties are useful for identifying what should go where and with what, to make it simpler to understand and to solve the equation properly. When we break things down to a simplified process, it is much easier to see how the real numbers are placed and why they are placed that way. Real numbers do not actually show the value of something real in the “real world”. For example, in mathematics if we write 0.5 we mean exactly half, but in the real world half may not be exactly half. In all reality, we use mathematics every single day, whether we consciously realize it or not. Math is the key subject that applies to our everyday lives in the “real world”.
Expression number one like terms are combined by adding coefficients, the removal of parentheses, and the use of commutative property of addition and multiplication. Expression number two has the use of quite a bit of distribution, combining like terms, and removal of parentheses. Expression number three like terms are combined by adding coefficients also. In this expression there is a temporary addition of...

...
Algebra 1
Schools in California now have higher expectations to make it necessary for students to take a Algebra 1 course in order to graduate from high school. This requirement issues that it will help students achieve higher expectations and great problem solving skills in future references. People like Mitchell Rosen a licensed family counselor who also disagrees with having Algebra 1 be a requirement for high schools. In one of Rosens articles “Finding X is not a factor of living,” he explains that algebra is not a reliable subject because it not used in the real world. Rosen argues that students should better life training skills in other subjects, students “need more [fundamental] training, not the fine-tuning.” Rosen argues that algebra can be discouraging to students and causes their self-esteem to decrease, also causing unnecessary stress for the student; algebra isn't required for most jobs in the real world; algebra has caused high school dropout rates to increase; due to low grades in algebra. Furthermore, algebra should not be a requirement in order to graduate from high school.
First and for most, If asked, most people would not say that they have personally never used algebraic problems outside of class room walls and isn't important to their professions, so Algebra 1 should not be a requirement in high...

...that there are many aspects of Algebra that the majority of people do not use on a daily basis. I think that this fact is what leads people to the false conclusion that Algebra is useless.
To better understand our topic, let’s define what we mean when we say “Algebra”. Webster’s dictionary defines Algebra as “a form of mathematics dealing with symbols and equations.” A guest in the mathematics forum on xpmath.com states that “…the truth is that Algebra is not much more than arithmetic expanded to the point where you don’t have to do trial and error to get an answer.” This guest goes on to explain that “…if you view it from that perspective, and overlook the outdated nature of some problems’ data, then you’ll recognize that indeed math deserves a place in your career; the more competent you can become with it, the better you’ll be able to competently manage you life.” I wholeheartedly agree with the preceding statement.
However, I’m not completely certain that math is THE MOST important subject we’ll ever learn; I believe that English quite important as well.
Math describes how everything in our environment works. A working knowledge of mathematics enables us to make accurate measurements and predictions. Since Algebra uses letters to represent numbers, it forces us to leap from concrete to abstract thinking. This “new thinking” method is, I believe, the reason Algebra...