
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 ...
... 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 ...
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 ...
... 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
... 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 ...
... 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
... 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. ...
... 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. ...
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 ...
... (*) 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 ...
Finding the Greatest Common Divisor by repeated
... Corollary: If a = GCD( b , c ) where b c , then then a divides (c b) ...
... Corollary: If a = GCD( b , c ) where b c , then then a divides (c b) ...
F08 Exam 1
... Please work only one problem per page, starting with the pages provided. Clearly label your answer. If a problem continues on a new page, clearly state this fact on both the old and the new pages. [1] Define a ring homomorphism. Define an ideal. Prove that the kernel of a ring homomorphism is an ide ...
... Please work only one problem per page, starting with the pages provided. Clearly label your answer. If a problem continues on a new page, clearly state this fact on both the old and the new pages. [1] Define a ring homomorphism. Define an ideal. Prove that the kernel of a ring homomorphism is an ide ...
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. ...
... 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. ...
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 ...
... 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
... 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 ...
... 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 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 ...
... 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 ...
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 ...
... 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
... 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 ...
... 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 ...