
Solving Quadratic Equations
... MATH 103 Class Activity – Solving Polynomial Equations (Accompanies Section 5.8) PART I – Quadratic Equations ...
... MATH 103 Class Activity – Solving Polynomial Equations (Accompanies Section 5.8) PART I – Quadratic Equations ...
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... Now we get to factoring a polynomial over Fp . Given a polynomial of degree f over Fp , it is enough to get one non-trivial1 factor of f. As we said in the last few lectures, the first thing to do is to check if f is square free. If it isn’t we can just return the square-free part of f as a factor a ...
... Now we get to factoring a polynomial over Fp . Given a polynomial of degree f over Fp , it is enough to get one non-trivial1 factor of f. As we said in the last few lectures, the first thing to do is to check if f is square free. If it isn’t we can just return the square-free part of f as a factor a ...
SOME HOMEWORK PROBLEMS Andrew Granville 1. Suppose that
... c) Show that if p1 = 1, p2 = 4, p3 = 8, p4 = 9, . . . is the sequence of powerful numbers then pn+2 − pn → ∞ as n → ∞. d) There are only finitely many powerful Fibonacci numbers. (Hint: Study solutions to x2 − 5y 2 = ±4). 4. Use the abc-conjecture to prove that there are only finitely many pairs of ...
... c) Show that if p1 = 1, p2 = 4, p3 = 8, p4 = 9, . . . is the sequence of powerful numbers then pn+2 − pn → ∞ as n → ∞. d) There are only finitely many powerful Fibonacci numbers. (Hint: Study solutions to x2 − 5y 2 = ±4). 4. Use the abc-conjecture to prove that there are only finitely many pairs of ...
18. Cyclotomic polynomials II
... any root ζ of Φ15 (x) = 0 (in some algebraic closure of , if one likes) is of degree equal to the degree of the polynomial Φ15 , namely ϕ(15) = ϕ(3)ϕ(5) = (3 − 1)(5 − 1) = 8. We already know that Φ3 and Φ5 are irreducible. And one notes that, given a primitive 15th root of unity ζ, η = ζ 3 is a prim ...
... any root ζ of Φ15 (x) = 0 (in some algebraic closure of , if one likes) is of degree equal to the degree of the polynomial Φ15 , namely ϕ(15) = ϕ(3)ϕ(5) = (3 − 1)(5 − 1) = 8. We already know that Φ3 and Φ5 are irreducible. And one notes that, given a primitive 15th root of unity ζ, η = ζ 3 is a prim ...
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.