Newsletters
... Here are some activities you and your student can do together: Items that are used together like hot dogs and buns or paper cups and paper plates, are often sold in different size packages. Look for examples of these in stores. Discuss the smallest number of packages of each item that you must buy ...
... Here are some activities you and your student can do together: Items that are used together like hot dogs and buns or paper cups and paper plates, are often sold in different size packages. Look for examples of these in stores. Discuss the smallest number of packages of each item that you must buy ...
Factoring Trinomials 2
... Step 4: Find the binomial factors of the trinomial. –8x3 + 2x2y + 3xy2 = –x(8x2 – 2xy – 3y2) = –x(2x + y)(4x – 3y) Step 5: Check by multiplication. –x(2x + y)(4x – 3y) = –x(8x2 – 6xy + 4xy – 3y2) = –x(8x2 – 2xy – 3y2) = –8x3 + 2x2y + 3xy2 = 2x2y – 8x3 + 3xy2 ...
... Step 4: Find the binomial factors of the trinomial. –8x3 + 2x2y + 3xy2 = –x(8x2 – 2xy – 3y2) = –x(2x + y)(4x – 3y) Step 5: Check by multiplication. –x(2x + y)(4x – 3y) = –x(8x2 – 6xy + 4xy – 3y2) = –x(8x2 – 2xy – 3y2) = –8x3 + 2x2y + 3xy2 = 2x2y – 8x3 + 3xy2 ...
I. Intro to Functions unit
... 2. Properties of quadratic functions a. Zeros: find them by: 1) factoring, 2) taking square roots (if appropriate) or 3) using the quadratic formula. b. Vertex: Three ways to find the x coordinate: average the zeros for factored form, b use x for standard form and just look at the equation for ve ...
... 2. Properties of quadratic functions a. Zeros: find them by: 1) factoring, 2) taking square roots (if appropriate) or 3) using the quadratic formula. b. Vertex: Three ways to find the x coordinate: average the zeros for factored form, b use x for standard form and just look at the equation for ve ...
12. Polynomials over UFDs
... [12.2] Let k be a field. Show that in the polynomial ring k[x, y] in two variables the ideal I = k[x, y] · x + k[x, y] · y is not principal. Suppose that there were a polynomial P (x, y) such that x = g(x, y) · P (x, y) for some polynomial g and y = h(x, y) · P (x, y) for some polynomial h. An intui ...
... [12.2] Let k be a field. Show that in the polynomial ring k[x, y] in two variables the ideal I = k[x, y] · x + k[x, y] · y is not principal. Suppose that there were a polynomial P (x, y) such that x = g(x, y) · P (x, y) for some polynomial g and y = h(x, y) · P (x, y) for some polynomial h. An intui ...
William Stallings, Cryptography and Network Security 3/e
... Polynomial Arithmetic in Zp • In the case of polynomial arithmetic performed on polynomials over a field, division is possible, but exact division might not be possible. To clarify, within a field, two elements a and b, the quotient a/b is also an element of the field. However, given a ring R that ...
... Polynomial Arithmetic in Zp • In the case of polynomial arithmetic performed on polynomials over a field, division is possible, but exact division might not be possible. To clarify, within a field, two elements a and b, the quotient a/b is also an element of the field. However, given a ring R that ...
Admission to Candidacy Examination in Algebra January 2011
... Note! You must show sufficient work to support your answer. Write your answers as legibly as you can on the blank sheets of paper provided. Use only one side of each sheet; start each problem on a new sheet of paper; and be sure to number your pages. Put your solution to problem 1 first, and then yo ...
... Note! You must show sufficient work to support your answer. Write your answers as legibly as you can on the blank sheets of paper provided. Use only one side of each sheet; start each problem on a new sheet of paper; and be sure to number your pages. Put your solution to problem 1 first, and then yo ...
PDF
... partial fractions? First one can take the highest power pν of a prime p which divides the denominator n. Then n = pν u, where gcd (u, pν ) = 1. Euclid’s algorithm gives some integers x and y such that 1 = xu+ypν . Dividing this equation by pν u gives the decomposition ...
... partial fractions? First one can take the highest power pν of a prime p which divides the denominator n. Then n = pν u, where gcd (u, pν ) = 1. Euclid’s algorithm gives some integers x and y such that 1 = xu+ypν . Dividing this equation by pν u gives the decomposition ...
Thinking Mathematically - Marquette University High School
... Step 3: SUBSTITUTE this expression into that variable in Equation B Equation B now becomes 7x – 2( 3x + 1 ) = - 4 Step 4: Solve for the remaining variable Step 5: Back-substitute this coordinate into Step 2 to find the other coordinate. (Or plug into any equation but step 2 is ...
... Step 3: SUBSTITUTE this expression into that variable in Equation B Equation B now becomes 7x – 2( 3x + 1 ) = - 4 Step 4: Solve for the remaining variable Step 5: Back-substitute this coordinate into Step 2 to find the other coordinate. (Or plug into any equation but step 2 is ...
MAT 0024
... Attendance: It is your responsibility to attend each lecture and keep records of assignments and other information delivered during class. Attendance will be recorded for administrative purposes and your instructor reserves the right to count it toward your final grade. Should this be the case, your ...
... Attendance: It is your responsibility to attend each lecture and keep records of assignments and other information delivered during class. Attendance will be recorded for administrative purposes and your instructor reserves the right to count it toward your final grade. Should this be the case, your ...
Multiplying binomials:
... Once you have that you can proceed. We will need factors of 6 with a difference of 1. The factors of 3 and 2 satisfy that and we will put the 3 with the minus sign because that is where the 1 st sign tells us to put the bigger #. ...
... Once you have that you can proceed. We will need factors of 6 with a difference of 1. The factors of 3 and 2 satisfy that and we will put the 3 with the minus sign because that is where the 1 st sign tells us to put the bigger #. ...
Factorization
In mathematics, factorization (also factorisation in some forms of British English) or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 × 5, and the polynomial x2 − 4 factors as (x − 2)(x + 2). In all cases, a product of simpler objects is obtained.The aim of factoring is usually to reduce something to “basic building blocks”, such as numbers to prime numbers, or polynomials to irreducible polynomials. Factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra. Viète's formulas relate the coefficients of a polynomial to its roots.The opposite of polynomial factorization is expansion, the multiplying together of polynomial factors to an “expanded” polynomial, written as just a sum of terms.Integer factorization for large integers appears to be a difficult problem. There is no known method to carry it out quickly. Its complexity is the basis of the assumed security of some public key cryptography algorithms, such as RSA.A matrix can also be factorized into a product of matrices of special types, for an application in which that form is convenient. One major example of this uses an orthogonal or unitary matrix, and a triangular matrix. There are different types: QR decomposition, LQ, QL, RQ, RZ.Another example is the factorization of a function as the composition of other functions having certain properties; for example, every function can be viewed as the composition of a surjective function with an injective function. This situation is generalized by factorization systems.