Addition of polynomials Multiplication of polynomials
... Example 7. Find the greatest common divisor of a(x) = 2x3 +x2 −2x−1 and b(x) = x3 −x2 +2x−2. Solution. We use the Euclidean Algorithm: first divide a(x) by b(x), then divide b(x) by the remainder, then divide the first remainder by the new remainder, and so on. The last non-zero remainder is the gre ...
... Example 7. Find the greatest common divisor of a(x) = 2x3 +x2 −2x−1 and b(x) = x3 −x2 +2x−2. Solution. We use the Euclidean Algorithm: first divide a(x) by b(x), then divide b(x) by the remainder, then divide the first remainder by the new remainder, and so on. The last non-zero remainder is the gre ...
HW3 Solution - UCSD VLSI CAD Lab
... 48ˆ[9]/52ˆ[9] (52ˆ[9] - 48ˆ[9])/(52ˆ[9]) (52ˆ[9] - 48ˆ[9] - 4*[9]*48ˆ[9-1])/(52ˆ[9]) C(48,[9])/C(52,[9]) (C(52,[9])-C(48,[9]))/C(52,[9]) (C(52,[9])-C(48,[9])-C(4,1)*C(48,[9-1]))/C(52,[9]) ...
... 48ˆ[9]/52ˆ[9] (52ˆ[9] - 48ˆ[9])/(52ˆ[9]) (52ˆ[9] - 48ˆ[9] - 4*[9]*48ˆ[9-1])/(52ˆ[9]) C(48,[9])/C(52,[9]) (C(52,[9])-C(48,[9]))/C(52,[9]) (C(52,[9])-C(48,[9])-C(4,1)*C(48,[9-1]))/C(52,[9]) ...
Here
... Example: Consider the polynomial x3 − 4x2 − 15x + 18. We know that 1 is a root of this polynomial because 13 − 4 · 12 − 15 · 1 + 18 = 0. So we should be able to write this polynomial in the form (x − 1)Q(x), where Q(x) is a polynomial. If we divide x3 −4x2 −15x+18 by x−1, we get x2 −3x−18 (with no r ...
... Example: Consider the polynomial x3 − 4x2 − 15x + 18. We know that 1 is a root of this polynomial because 13 − 4 · 12 − 15 · 1 + 18 = 0. So we should be able to write this polynomial in the form (x − 1)Q(x), where Q(x) is a polynomial. If we divide x3 −4x2 −15x+18 by x−1, we get x2 −3x−18 (with no r ...
pdf-file
... (iii) If there is no constant term, then x is a common factor. For example, x2 + 4x = x(x + 4). ...
... (iii) If there is no constant term, then x is a common factor. For example, x2 + 4x = x(x + 4). ...
3.3 Introduction to Polynomials
... Polynomials which are larger than 3 terms are simply referred to by the number of terms they have, that is, “a 4-term polynomial, a 5-term polynomial, etc”. Now we must be careful with our definition above, the fact that it has “no division” does not mean we can’t divide two polynomials. It just mea ...
... Polynomials which are larger than 3 terms are simply referred to by the number of terms they have, that is, “a 4-term polynomial, a 5-term polynomial, etc”. Now we must be careful with our definition above, the fact that it has “no division” does not mean we can’t divide two polynomials. It just mea ...
Applications of Fibonacci Numbers
... "convoluted" algebraic methods. Yet the presence of both Fibonacci numbers and binomial coefficients demands a combinatorial explanation. Beginning with our proof of Identity 1, we provide simple, combinatorial arguments for many fibanomid identities - identities that combine (generalized) Fibonacci ...
... "convoluted" algebraic methods. Yet the presence of both Fibonacci numbers and binomial coefficients demands a combinatorial explanation. Beginning with our proof of Identity 1, we provide simple, combinatorial arguments for many fibanomid identities - identities that combine (generalized) Fibonacci ...
Complex Polynomial Identities
... form a + bi and a – bi. Both numbers contain an imaginary part, but multiplying them produces a value that is wholly real. Therefore, the complex conjugate of a + bi is a – bi, and vice versa. • The sum of two squares can be rewritten as the product of complex conjugates: a2 + b2 = (a + bi)(a – bi), ...
... form a + bi and a – bi. Both numbers contain an imaginary part, but multiplying them produces a value that is wholly real. Therefore, the complex conjugate of a + bi is a – bi, and vice versa. • The sum of two squares can be rewritten as the product of complex conjugates: a2 + b2 = (a + bi)(a – bi), ...
PDF
... Since d was the first place where this sort of wraparound happened, d is the least such number such that pd = 1 mod r which implies that d is the order of p modulo r. This is denoted by d = ordr (p). Thus, the degree of the the minimum polynomial of ζ over Fp for any primitive r-th root is ordr (p). ...
... Since d was the first place where this sort of wraparound happened, d is the least such number such that pd = 1 mod r which implies that d is the order of p modulo r. This is denoted by d = ordr (p). Thus, the degree of the the minimum polynomial of ζ over Fp for any primitive r-th root is ordr (p). ...
Algebra Notes
... a field into a bigger field: start with a field F , then first enlarge it into the polynomial ring with coefficients in that field F [x], then take the quotient of this ring by one of its ideals I. If I is a maximal ideal, then F [x]/I will be a new field which contains F as a subfield. To begin stu ...
... a field into a bigger field: start with a field F , then first enlarge it into the polynomial ring with coefficients in that field F [x], then take the quotient of this ring by one of its ideals I. If I is a maximal ideal, then F [x]/I will be a new field which contains F as a subfield. To begin stu ...
Lesson4 - Purdue Math
... To multiply two general polynomials with 2 or more terms: Use a variation of FOIL. Distribute each term of the first polynomial to every term of the second polynomial and combine the results. Ex 6: Multiply and write product in simplified form. ...
... To multiply two general polynomials with 2 or more terms: Use a variation of FOIL. Distribute each term of the first polynomial to every term of the second polynomial and combine the results. Ex 6: Multiply and write product in simplified form. ...
Appendix B The Least Squares Regression Line The least squares
... r =correlation coefficient, sy = Standard deviation of y, sx = Standard deviation of x Thus the equation for the least squares line can be symbolically represented as sy sy sy yˆ = r ( x) + ( y − r ( x )) Or an alternative form of yˆ = r ( x − x ) + y . sx sx sx FOCUS ON MATHEMATICS Q33. The least s ...
... r =correlation coefficient, sy = Standard deviation of y, sx = Standard deviation of x Thus the equation for the least squares line can be symbolically represented as sy sy sy yˆ = r ( x) + ( y − r ( x )) Or an alternative form of yˆ = r ( x − x ) + y . sx sx sx FOCUS ON MATHEMATICS Q33. The least s ...
WHAT IS A POLYNOMIAL? 1. A Construction of the Complex
... It is clear how to add such sequences and how to scale such a sequence by an element of R, (an ) + (bn ) = (an + bn ), r(an ) = (ran ). Define the product of two such sequences to be (an )(bn ) = (cn ) ...
... It is clear how to add such sequences and how to scale such a sequence by an element of R, (an ) + (bn ) = (an + bn ), r(an ) = (ran ). Define the product of two such sequences to be (an )(bn ) = (cn ) ...
Document
... written as a product of polynomials of lesser degree using only integer coefficients and constants and if the only common factors of its terms are _1_ and _1_. Example 16x2 4x + 8 _is not_ a prime polynomial because _4_ is a common factor of all its terms. Definition A polynomial is factored comp ...
... written as a product of polynomials of lesser degree using only integer coefficients and constants and if the only common factors of its terms are _1_ and _1_. Example 16x2 4x + 8 _is not_ a prime polynomial because _4_ is a common factor of all its terms. Definition A polynomial is factored comp ...
17-Distribution Combine Like terms
... The leading term is 3x6. There are five terms The leading term is 3x6 the exponent of the first term is 6. The leading coefficient is 6. This is a sixth-degree polynomial. ...
... The leading term is 3x6. There are five terms The leading term is 3x6 the exponent of the first term is 6. The leading coefficient is 6. This is a sixth-degree polynomial. ...
Solutions
... In this particular case, there are 8 sides, so the answer would be 12. 2) A regular tetrahedron is a triangular pyramid whose faces are all equilateral triangles. How many distinguishable ways can we paint the four face of a regular tetrahedron with red, blue, green, and orange paint such that no tw ...
... In this particular case, there are 8 sides, so the answer would be 12. 2) A regular tetrahedron is a triangular pyramid whose faces are all equilateral triangles. How many distinguishable ways can we paint the four face of a regular tetrahedron with red, blue, green, and orange paint such that no tw ...
PDF
... The theorem implies, similarly as in the ring Z of the integers, that one can perform in K[X] a Euclid’s algorithm which yields a greatest common divisor of two polynomials. Performing several consecutive Euclid’s algorithms one obtains a gcd of many polynomials; such a gcd is always in the same pol ...
... The theorem implies, similarly as in the ring Z of the integers, that one can perform in K[X] a Euclid’s algorithm which yields a greatest common divisor of two polynomials. Performing several consecutive Euclid’s algorithms one obtains a gcd of many polynomials; such a gcd is always in the same pol ...
PDF
... Elements of R[X] are called polynomials in the indeterminate X with coefficients in R. The ring elements a0 , . . . , aN are called coefficients of the polynomial, and the degree of a polynomial is the largest natural number N for which aN 6= 0, if such an N exists. When a polynomial has all of its ...
... Elements of R[X] are called polynomials in the indeterminate X with coefficients in R. The ring elements a0 , . . . , aN are called coefficients of the polynomial, and the degree of a polynomial is the largest natural number N for which aN 6= 0, if such an N exists. When a polynomial has all of its ...
Math 403A assignment 7. Due Friday, March 8, 2013. Chapter 12
... A nonconstant polynomial f (x) ∈ Q[x] is primitive if it has integer coefficients, the g.c.d. of its coefficients is 1, and its leading coefficient is positive. Gauss. The product of primitive polynomials is primitive. Gauss’s lemma says that the primitive polynomials form a multiplicative subset of ...
... A nonconstant polynomial f (x) ∈ Q[x] is primitive if it has integer coefficients, the g.c.d. of its coefficients is 1, and its leading coefficient is positive. Gauss. The product of primitive polynomials is primitive. Gauss’s lemma says that the primitive polynomials form a multiplicative subset of ...
Unit 2: Polynomials And Factoring
... Unit 2: Polynomials And Factoring Content Area: Course(s): Time Period: Length: Status: ...
... Unit 2: Polynomials And Factoring Content Area: Course(s): Time Period: Length: Status: ...
H8
... 1. Let a(x) = (x − 2)2 and b(x) = (x − 3)2 in R[x]. (a) Find polynomials u(x) and v(x) in R[x] so that a(x)u(x) + b(x)v(x) = 1. (b) Find reconstruction polynomials c1 (x), c2 (x) ∈ R[x] so that given any f1 (x) and f2 (x) in R[x] the polynomial f (x) = c1 (x)f1 (x) + c2 (x)f2 (x) satisfies f (x) ≡ f ...
... 1. Let a(x) = (x − 2)2 and b(x) = (x − 3)2 in R[x]. (a) Find polynomials u(x) and v(x) in R[x] so that a(x)u(x) + b(x)v(x) = 1. (b) Find reconstruction polynomials c1 (x), c2 (x) ∈ R[x] so that given any f1 (x) and f2 (x) in R[x] the polynomial f (x) = c1 (x)f1 (x) + c2 (x)f2 (x) satisfies f (x) ≡ f ...
Quadratic forms - University of Toronto
... Notice the dual role played by v here, as the argument in the polynomial and as the constant coefficient of the polynomial. This is a little subtle, but the student who becomes proficient at mathematics will have to become accustomed to such switches in standpoint. A more general version of this res ...
... Notice the dual role played by v here, as the argument in the polynomial and as the constant coefficient of the polynomial. This is a little subtle, but the student who becomes proficient at mathematics will have to become accustomed to such switches in standpoint. A more general version of this res ...
Question Set 2 - University of Toronto
... Notice the dual role played by v here, as the argument in the polynomial and as the constant coefficient of the polynomial. This is a little subtle, but the student who becomes proficient at mathematics will have to become accustomed to such switches in standpoint. A more general version of this res ...
... Notice the dual role played by v here, as the argument in the polynomial and as the constant coefficient of the polynomial. This is a little subtle, but the student who becomes proficient at mathematics will have to become accustomed to such switches in standpoint. A more general version of this res ...
Alg 2 Chapter 1 Jeopardy
... Give the domain of the following. Mapping the face numbers on a clock to their corresponding number of seconds ...
... Give the domain of the following. Mapping the face numbers on a clock to their corresponding number of seconds ...