18.781 Problem Set 3
18.781 Problem Set 3

Solution - Stony Brook Mathematics
Solution - Stony Brook Mathematics

... than x (since x/2 < x by the multiplicative law for inequalities), a contradiction. Therefore, there does not exist a smallest positive real number. Problem 4. Use induction to prove Bernouilli’s inequality : (1 + x)n ≥ 1 + nx for all non-negative integers n and real numbers x > −1. Sol. Fix some re ...
8.4 Solve Linear Systems by Elimination Using Multiplication
8.4 Solve Linear Systems by Elimination Using Multiplication

methods of proofs
methods of proofs

Cubic Formula
Cubic Formula

REAL NUMBERS What Are Real Numbers?
REAL NUMBERS What Are Real Numbers?

Number theory and proof techniques
Number theory and proof techniques

The ring of evenly weighted points on the projective line
The ring of evenly weighted points on the projective line

A Case of Depth-3 Identity Testing, Sparse Factorization and Duality
A Case of Depth-3 Identity Testing, Sparse Factorization and Duality

Chapter 1 Notes
Chapter 1 Notes

PMAT 527/627 Practice Midterm
PMAT 527/627 Practice Midterm

A matroid analogue of a theorem of Brooks for graphs
A matroid analogue of a theorem of Brooks for graphs

1. Introduction 2. Curry algebras
1. Introduction 2. Curry algebras

here
here

CSC 2500 Computer Organization
CSC 2500 Computer Organization

Algebra 1 Mod 1 Review Worksheet I. Graphs Consider the graph
Algebra 1 Mod 1 Review Worksheet I. Graphs Consider the graph

Quadratics
Quadratics

Existence and uniqueness of Haar integrals
Existence and uniqueness of Haar integrals

Chapter 1 – ALGEBRA REVIEW
Chapter 1 – ALGEBRA REVIEW

... 4.6 Factor a polynomial using only linear factors. 4.6 Determine the equation of a polynomial given its zeros or graph. 5.1 Define and apply rational exponents. 5.1 Simplify expressions containing radicals or rational exponents. 5.2 Graph and identify transformations of exponential functions. 5.3 U ...
P - Coastal Bend College
P - Coastal Bend College

Algebra Brain Summary
Algebra Brain Summary

Solving Quadratic Equations by Finding Square Roots
Solving Quadratic Equations by Finding Square Roots

Math 3: Unit 1 – Reasoning and Proof Inductive, Deductive
Math 3: Unit 1 – Reasoning and Proof Inductive, Deductive

31 Semisimple Modules and the radical
31 Semisimple Modules and the radical

2 is irreducible in Q[ √ 2]
2 is irreducible in Q[ √ 2]

... to 7 so it cannot be isomorphic to Q[ 7]. • Exercises 4.1: 9, 14(a)(c). I went over these ones or very similar ones in class so I’m not going to do them here. • Exercises 3.3: 2(a)(c)(e), 4(a). 2(a) Is x3 + 4x2 − 3x + 5 irreducible in Q[x]? Its irreducible if and only if it has no rational roots whi ...

Fundamental theorem of algebra

The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with an imaginary part equal to zero.Equivalently (by definition), the theorem states that the field of complex numbers is algebraically closed.The theorem is also stated as follows: every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n roots. The equivalence of the two statements can be proven through the use of successive polynomial division.In spite of its name, there is no purely algebraic proof of the theorem, since any proof must use the completeness of the reals (or some other equivalent formulation of completeness), which is not an algebraic concept. Additionally, it is not fundamental for modern algebra; its name was given at a time when the study of algebra was mainly concerned with the solutions of polynomial equations with real or complex coefficients.