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...

...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...

...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....

...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...

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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...

...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. See...

...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...

...Accelerated Coordinate Algebra / Analytic Geometry Part A
Dr. Khan, Ph.D., Fall 2012
ekhan@marietta-city.k12.ga.us
WHY ARE YOU TAKING THIS COURSE?
All Georgia high school students are required to take four years of mathematics. Taking Accelerated Coordinate Algebra / Analytic Geometry Part A is comparable to taking the typical ninth grade course, Coordinate Algebra AND the first half of the tenth grade course, Analytic Geometry. The reason...