Olympiad Algebra and Analysis
Read this topicExplanations 8
- Polynomials Definition Division Algorithm Remainder / Factor Theorem ** Rational Root Theorem** ** Identical Polynomials** ** More Exercises**
By Farrell Eldrian Wu
- Factorizations Definitions and Methods Quadratics - Quadratic Formula (factoring linear out) Cubics and Higher-degree Polynomials (factoring linear out ) Factor Theorem and its …
By Farrell Eldrian Wu
- Hello! This is sort of a Tricki-style experiment, but not exactly... Please/feel free to contribute any ("algebraic") heuristic/trick/strategy here, no matter what level, or how…
By Victor Wang, Victor Wang
- Common Sets in Algebra (Part 2) In the last part of the article, I explained about the set of natural numbers, integers, rational numbers, real numbers, and complex numbers. The…
By Zi Song Yeoh
- This is a stub article; the stuff on transformations and completing the square should be expanded. The goal of this post is to show (algebraically) that all quadratics have the …
By Victor Wang
- The quadratic formula gives the roots of a quadratic, at least when the discriminant is nonnegative. But some quadratics have no real roots, so in order to "have solutions" we n…
By Victor Wang
- Stub explanation The quadratic discriminant b2−4acb^2-4acb2−4ac of a real quadratic P(x)=ax2+bx+cP(x) = ax^2+bx+cP(x)=ax2+bx+c has a nice interpretation in terms of the roots r1…
By Victor Wang
- Recall that the complex numbers are numbers of the form a+bia+bia+bi where a,b∈Ra, b \in \mathbb{R}a,b∈R, i.e. aaa and bbb are real numbers, and the symbol iii satisfies the equ…
By Mendel Cyprys
Problems 6
- Check the boxes corresponding to algebraic integers:
- Consider a set satisfying the below properties: i) It is a subset of the set of rational numbers. ii) −200-200−200 is in the set. iii) 0.1234512345....0.1234512345....0.12345123…
- If A,B,CA,B,CA,B,C are positive reals such that An≥BnCA^n \ge \frac{B^n}{C}An≥CBn for n=1,2,3,…n=1,2,3,\dotsn=1,2,3,…, then what can we conclude?
- Can we find a sequence of reals ai≠0a_i\neq0ai=0 such that each polynomial pn(x)=a0+a1x+⋯+anxnp_n(x) = a_0 + a_1 x + \dots + a_n x^npn(x)=a0+a1x+⋯+anxn has all its roots r…
- What is the asymptotic growth of I(n)I(n)I(n) as n→∞n\to\inftyn→∞? \\ [Source: Most recently communicated by Mark Sellke]
- If x⃗,y⃗∈Rn\vec{x}, \vec{y} \in \mathbb{R}^nx,y∈Rn are such that Q(x⃗,x⃗)>0Q( \vec{x}, \vec{x}) > 0Q(x,x)>0 and Q(y⃗,y⃗)>0Q( \vec{y}, \vec{y}) > 0Q(y,y)>0, is it possible tha…