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Adding and Subtracting Polynomials
Adding and Subtracting Polynomials

... The degreeThe degree The degree The degree a polynomial is the degree of thisofterm is of 2x2 isof2 any 4. with the constant for the term highest ispower. In ...
Automatic Geometric Theorem Proving: Turning Euclidean
Automatic Geometric Theorem Proving: Turning Euclidean

Elementary Properties of the Integers
Elementary Properties of the Integers

... Theorem (2-2). If a and b are integers, not both zero, then gcd(a, b) exists and is unique. Note that this result is trivial if either a or b is zero. Also note that changing the sign of a or b or swapping a for b does not change the gcd. Without loss of generality, we may thus assume that 0 < b ≤ a ...
Appendix on Algebra
Appendix on Algebra

... and distinguished elements 0 and 1 (the additive and multiplicative identities). Every number a ∈ F has an additive inverse −a and every non-zero number a ∈ F× := F − {0} has a multiplicative inverse a−1 =: a1 . Addition and multiplication are commutative and associative and multiplication distribut ...
x - ckw
x - ckw

... 5.I. Complex Vector Spaces A complex vector space is a linear space with complex numbers as scalars, i.e., the scalar multiplication is over C, the complex number field. All n-D complex vector spaces are isomorphic to Cn. ...
Lesson 2 – Multiplying a polynomial by a monomial
Lesson 2 – Multiplying a polynomial by a monomial

hw2.pdf
hw2.pdf

... (*) 14. (Gallian, p.57, #34) Prove that if G is a group and a, b ∈ G then (ab)2 = a2 b2 if and only if ab = ba . 15. Give an example of a group G and a, b ∈ G so that (ab)4 = a4 b4 , but ab 6= ba. [Hint: Problem #13 might help? Slightly bigger challenge: try the same thing with the 4’s replaced by 3 ...
Algebraic Statistics
Algebraic Statistics

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Finding the Greatest Common Divisor by repeated

... Corollary: If a = GCD( b , c ) where b  c , then then a divides (c  b) ...
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Further Number Theory
Further Number Theory

... The Euclidean Algorithm involves using repeated applications of the division algorithm to find the greatest common divisor of two integers. It is best shown by example. Examples ① Find the greatest common divisor of 146 & 14. ...
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Ex. 3x5 + 6x4 - 2x3 + x2 + 7x - 6 degree: coefficients: leading

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2.7 Apply the Fundamental Theorem of Algebra

Multiplying Polynomials
Multiplying Polynomials

... Multiplying Polynomials • To multiply polynomials together each term of the first polynomial must be multiplied by each term of the second polynomial. • There are several methods for multiplying polynomials. • The three methods we will be using are the box method, using the memory device FOIL, and ...
lesson - Effingham County Schools
lesson - Effingham County Schools

... 1. Rewrite the polynomial in standard form. Then identify the leading coefficient, degree, and number of terms. Name/classify the polynomial. x 2  3  2x 5  7x 4  12x ...
Algebra - Phillips9math
Algebra - Phillips9math

... Algebra Algebra is another one of those ‘bad’ math words that most people dislike. However, the fact is that algebra is nothing more than working with unknown numbers. We call these numbers ‘variables’. Here is some more vocabulary to get us on our way in our algebra unit: Expand ...
POLYNOMIALS 1. Polynomial Rings Let R be a commutative ring
POLYNOMIALS 1. Polynomial Rings Let R be a commutative ring

Problems - NIU Math
Problems - NIU Math

... with the other examples in the text. The axioms of field are the ones we need to work with polynomials and matrices, so these are the primary examples in the section. The remainder theorem (Theorem 4.1.9) is a special case of the division algorithm (Theorem 4.2.1). Since the proof can be given much ...
simple algebra
simple algebra

... Addition – coefficient by coefficient addition – the coefficients remain in the same field Multiplication by a scalar – multiply the coefficients by the scalar Multiplication of two polynomials – the high-school method Division – the high-school method – note that A(X)/B(Z) is really A(X) mod B(X) a ...
Polynomials--found poetry
Polynomials--found poetry

gcd( 0,6)
gcd( 0,6)

Section 2.1
Section 2.1

Project 1 - cs.rochester.edu
Project 1 - cs.rochester.edu

x x xx x x = = = 2 5 2(5) 10 10 x x x x x x = = = 3 5 7 3 3
x x xx x x = = = 2 5 2(5) 10 10 x x x x x x = = = 3 5 7 3 3

< 1 ... 40 41 42 43 44 45 >

Polynomial greatest common divisor

In algebra, the greatest common divisor (frequently abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a factor of both the two original polynomials. This concept is analogous to the greatest common divisor of two integers.In the important case of univariate polynomials over a field the polynomial GCD may be computed, like for the integer GCD, by Euclid's algorithm using long division. The polynomial GCD is defined only up to the multiplication by an invertible constant.The similarity between the integer GCD and the polynomial GCD allows us to extend to univariate polynomials all the properties that may be deduced from Euclid's algorithm and Euclidean division. Moreover, the polynomial GCD has specific properties that make it a fundamental notion in various areas of algebra. Typically, the roots of the GCD of two polynomials are the common roots of the two polynomials, and this allows to get information on the roots without computing them. For example, the multiple roots of a polynomial are the roots of the GCD of the polynomial and its derivative, and further GCD computations allow to compute the square-free factorization of the polynomial, which provides polynomials whose roots are the roots of a given multiplicity.The greatest common divisor may be defined and exists, more generally, for multivariate polynomials over a field or the ring of integers, and also over a unique factorization domain. There exist algorithms to compute them as soon as one has a GCD algorithm in the ring of coefficients. These algorithms proceed by a recursion on the number of variables to reduce the problem to a variant of Euclid's algorithm. They are a fundamental tool in computer algebra, because computer algebra systems use them systematically to simplify fractions. Conversely, most of the modern theory of polynomial GCD has been developed to satisfy the need of efficiency of computer algebra systems.
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