What is Factorization: Definition and 160 Discussions
In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. For example, 3 × 5 is a factorization of the integer 15, and (x – 2)(x + 2) is a factorization of the polynomial x2 – 4.
Factorization is not usually considered meaningful within number systems possessing division, such as the real or complex numbers, since any
x
{\displaystyle x}
can be trivially written as
(
x
y
)
×
(
1
/
y
)
{\displaystyle (xy)\times (1/y)}
whenever
y
{\displaystyle y}
is not zero. However, a meaningful factorization for a rational number or a rational function can be obtained by writing it in lowest terms and separately factoring its numerator and denominator.
Factorization was first considered by ancient Greek mathematicians in the case of integers. They proved the fundamental theorem of arithmetic, which asserts that every positive integer may be factored into a product of prime numbers, which cannot be further factored into integers greater than 1. Moreover, this factorization is unique up to the order of the factors. Although integer factorization is a sort of inverse to multiplication, it is much more difficult algorithmically, a fact which is exploited in the RSA cryptosystem to implement public-key cryptography.
Polynomial factorization has also been studied for centuries. In elementary algebra, factoring a polynomial reduces the problem of finding its roots to finding the roots of the factors. Polynomials with coefficients in the integers or in a field possess the unique factorization property, a version of the fundamental theorem of arithmetic with prime numbers replaced by irreducible polynomials. In particular, a univariate polynomial with complex coefficients admits a unique (up to ordering) factorization into linear polynomials: this is a version of the fundamental theorem of algebra. In this case, the factorization can be done with root-finding algorithms. The case of polynomials with integer coefficients is fundamental for computer algebra. There are efficient computer algorithms for computing (complete) factorizations within the ring of polynomials with rational number coefficients (see factorization of polynomials).
A commutative ring possessing the unique factorization property is called a unique factorization domain. There are number systems, such as certain rings of algebraic integers, which are not unique factorization domains. However, rings of algebraic integers satisfy the weaker property of Dedekind domains: ideals factor uniquely into prime ideals.
Factorization may also refer to more general decompositions of a mathematical object into the product of smaller or simpler objects. For example, every function may be factored into the composition of a surjective function with an injective function. Matrices possess many kinds of matrix factorizations. For example, every matrix has a unique LUP factorization as a product of a lower triangular matrix L with all diagonal entries equal to one, an upper triangular matrix U, and a permutation matrix P; this is a matrix formulation of Gaussian elimination.
(expression given to be proven)
check for p(1)... 2=2
substitute (n+n) to
And here is the problem, I just can't find a way to continue solving this problem
## 1234=2\cdot 617 ##
## 10140=2\cdot 2\cdot 3\cdot 5\cdot 13\cdot 13 ##
## 36000=2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 3\cdot 3\cdot 5\cdot 5\cdot 5\cdot ##
Are the answers above correct? Or do I need to put them in canonical form as below?
## 1234=2\cdot 617 ##
## 10140=2^{2}\cdot 3\cdot 5\cdot...
I have been tasked with calculating amplitudes of a B meson decaying to a photon and lepton/lepton anti-neutrino pair ,upto one loop and have pretty much never seen this thing before. I will ask my questions along the way as I describe what I am doing.
This factorization theorem seems to go thus...
Build the least common multiple of A, B, and C
Then write the prime factorization of the least common multiple of A, B, and C.
$A = 2 \cdot 3^2 \cdot 7 \cdot 13 \cdot 23^8$
$B = 2 3^5 \cdot 5^9 \cdot 13$
$C = 2 \cdot 5 \cdot 11^8 \cdot 13^3$
$\boxed{?}$
ok this only has a single answer...
Build the least common multiple of A and B
a. write the prime factorization of the least common multiple of A and B.
$A=2\cdot 3^2\cdot 5\cdot 7^3\cdot 11^3\cdot 13^2$
$B= 3^2 \cdot 5 \cdot 7^2 \cdot 11^2$
$\dfrac{2\cdot \cancel{3^2}\cdot \cancel{5\cdot 7^2} 7\cdot \cancel{11^2} 11\cdot...
I am going to give up a bit more on the given problem. We start with polynomial ## x^27 -x ## over GF(3)[x] and we factorize it using a well known theorem it turns out it factorises into the product of monic polynomials of degree 1 and 3, 11 of them all together.
We then choose one of those...
If you go to "The Abel Prize Interview 2016 with Andrew Wiles" on YouTube, there is a statement made by Andrew Wiles beginning at about 4:10 and ending about 4:54 where he mentions there are some new number systems possible where the fundamental theorem of arithmetic does not hold. I don't...
Hello all, I have a problem related to LU Factorization with my work following it. Would anyone be willing to provide feedback on if my work is a correct approach/answer and help if it needs more work? Thanks in advance.
Problem:
Work:
$\tiny{apc.1.1.13}$
The number 1,001 is the product of the primes 7, 11, and 13
Knowing this,
What is the prime factorization of 30,030?
a, ${3 \cdot 7\cdot 10\cdot 13}$
b. ${30\cdot 7\cdot 11\cdot 13}$
c. ${2 \cdot 5\cdot 7\cdot 11\cdot 13}$
d. ${3\cdot 7\cdot 10\cdot 11\cdot 13}$
e...
What's the correct command for finding an LU factorization of a 3x3 and 4x4 matrix on Ti-89 graphing calculator? I'm trying to find the correct answers and verify/check my answers for Linear Algebra problems.
I know that the prime factorization theorem predicts that a prime number raised to an integer power will never be equal to another prime number raised to a different power. But does this apply to real number powers? For example, suppose there is a prime number raised to some real value, could...
I don't understand proof of uniqueness theorem for polynomial factorization, as described in Stewart's "Galois Theory", Theorem 3.16, p. 38.
"For any subfield K of C, factorization of polynomials over K into irreducible polynomials in unique up to constant factors and the order in which the...
Homework Statement
Define {x \choose n}=\frac{x(x-1)(x-2)...(x-n+1)}{n!} for positive integer n. For what values of positive integers n and m is g(x)={{{x+1} \choose n} \choose {m}}-{{{x} \choose n} \choose {m}} a factor of f(x)={{{x+1} \choose n} \choose {m}}?
Homework Equations
The idea...
I am reading The Basics of Abstract Algebra by Paul E. Bland ...
I am focused on Section 7.2 Euclidean, Principal Ideal, Unique Factorization Domains ... ...
I need help with the proof of Theorem 7.2.20 ... ... Theorem 7.2.20 and its proof reads as...
Homework Statement
Looking to factor ##-2x^3-3## and having an issue. To my understanding, the Fundamental Theorem of Algebra tells us that it is at least theoretically possible to factor any polynomial of degree n.
Homework EquationsThe Attempt at a Solution
So my first step to factor this...
Hello,
I'm looking for the non-negative matrix factorization (NNMF) source code. I checked several linear algebra libraries (e.g., LaPack, mkl), but it seems that this subroutine is not available. Does anyone know where I can find this source code...
So I am kind of lost... I don't really know how to ask this.
Project Euler is a website that hosts multiple programming contests and I am interested in this problem
https://projecteuler.net/problem=608
but my question isn't truly about this problem but a more solution.
I know that the Divisor...
Hi All,
I am doing some FA on SPSS. I entered 9 columns from an Excel file into
SPSS. I am just not sure of how the order of the initial columns corresponds to the order
of the components that have been extracted; specifically, in the "total Variance Explained" section of the outputs. Would...
Homework Statement
The book wants me to use direct proof.
if p is a prime and k is an integer for which 0 < k < p, then p divides ##\left( \frac p k \right)##
Homework Equations
##\left( \frac p k \right) = \frac {p!} {k!(p-k)!}##
The Attempt at a Solution
the fraction line in ##\left( \frac...
Hi,
Can anyone please tell me any easy way of prime factorizing 5-digit composite numbers from 10,000 to 99,999 with little writing or mentally?
Thanks.
Homework Statement
Hello, pretty back to basics with this one. How this came about was I am finding the eigenvalues for a given matrix and after forming the characteristic polynomial of the matrix i get this.
x^3 - 2x^2 -15x +36Homework Equations
Using the rational root theorem i came to the...
mabe someone can help me with this code:
i have this code, basically first i factorize for example the number 28 to: [2,2,7] and then i make a list of prime numbers and find the index of each factor in that list, so 2 is prime number with index 0 and 7 prime number with index 2 so it ends up...
This might be a dumb question, but I am wondering, given the equation ##A\vec{x} - 7\vec{x} = \vec{0}##, the factorization ##(A - 7I)\vec{x} = \vec{0}## is correct rather than the factorization ##(A - 7)\vec{x} = \vec{0}##. It seems that I can discribute just fine to get the equation we had...
Hi, quick question with A being the lowering operator and A† the raising operator for a QHO
(A A† - 1 + 1/2) ħω [Aψ] = A (A† A - 1 + 1/2) ħω ψ
By taking out a factor of A. Why has the ordering of A A† swapped around? I would have thought taking out a factor of A would leave it as
A (A† - 1 +...
In interpretations without natural factorizations, the cat won't just be dead or alive. It won't even be a cat. So let's say the cat is isolated in a box totally shielded from any decoherence from any environment.. and the any factorization between system and environment inside the box is...
ok so is there a function that exists (for all intents and purposes let's call it G(x,y) )where
x= a^2*b^4*c
y=a^4*b^2*d
G(x,y) = a^2*b^4
basically gcd, but the exponents match those of the common prime factors of the first input (x)
********
equally useful would be a function where the...
hey so
if you are taking a floor function of a fraction >1, is there any way to predict anything about it's factorization?
what about when the numerator is a factorial and the denominator is made up of factors that divide said factorial but to larger exponents then those that divide the...
Dear Everyone,
I have some trouble with LU-Factorization. The problem work is below:
Original Matrix
First row- 3,0,1
2nd row- 6,1,1
3rd row- -3,1,0
Elementary Matrix 1
1st- 1,0,0
2nd- 0,1,0
3rd- 0,0,1
Switch R1 to R3
1st- 0,0,1
2nd- 0,1,0
3rd- 1,0,0...
It is a problem found on IB math SL book:The two perpendicular sides of a right-angled triangle have lengths x+2 and 5x-3, the hypotenuse has length 4x+1, Find X.
In the answer section, it says X is either 2/5 or 3.
I tried a lot, -> i arrived to 10x^2 - 18x -12= 0
Later i used quadractic...
I am reading W. Keith Nicholson's book: Introduction to Abstract Algebra (Third Edition) ...
I am focused on Section 4.2:Factorization of Polynomials over a Field.
I need some help with the proof of Part 1 of Theorem 12 on page 218
The relevant text from Nicholson's book is as...
I am reading W. Keith Nicholson's book: Introduction to Abstract Algebra (Third Edition) ...
I am focused on Section 4.2:Factorization of Polynomials over a Filed.
I need some help with Example 10 on page 215 ...
The relevant text from Nicholson's book is as follows:In the above text, we read...
Homework Statement
Consider an invertible n x n matrix A. Can you write A as A=LQ, where L is a lower triangular matrix and Q is orthogonal? Hint: Consider the QR factorization of #A^T#.Homework Equations
For QR factorization, Q is orthogonal and R is upper triangular.
The Attempt at a...
I'm trying to figure out what this quote means and what our strategy is just looking at the matrix what kind of permutations we need to do.
Quotes: "The first permutation step is trivial (since the pivot element 10 is already the largest). The corresponding permutation matrix is the identity...
Here is a direct quote from my textbook:
If R is a commutative ring, we say that a polynomial d in R[x] is a divisor of f in R[x] if f = qd for some q in F[x].
My question is did they mean to put q in F[x}? q isn't in R[x]? They didn't mention F[x] before this, is F[x] the field of all...
Let $R$ be an integral domain. Suppose that $R_1$ and $R_2$ are proper subrings of $R$ and that both $R_1$ and $R_2$ are unique factorization domains (UFDs). Let $R_3$ be the subring of $R$ that is generated by $R_1$ and $R_2$. Is $R_3$ necessarily a UFD? (The subring generated by two subrings...
Can anyone tell me how to solve the following limit by factorization method
$\lim{{x}\to{5}} \frac{x^3 + 3x^2 - 6x + 2}{ x^3 + 3x^2 - 3x - 1}$?Please tell me how to factorize such big equation?
Hi, I am new to factorization. Would someone please solve these two equations and explain step by step what was done. thanks1. (x-y)^2 - (x-z)^2 =
2. (5x+2)^2 - (3x-4)^2 =
This lemma the book states, I can't make sense of it.
Lemma: If a,b\in Z and b > 0, there exist q,r \in Z such that a = qb + r with 0 \leq r < b.
Proof: Consider the set of all integers of the form a-xb with x \in Z. This set includes positive elements. Let r = a - qb be the least...
I am reading Chapter 2: Commutative Rings in Joseph Rotman's book, Advanced Modern Algebra (Second Edition).
I am currently focussed on Theorem 2.60 (Unique Factorization) [pages 111 - 112].
I need help to understand Rotman's use of induction (or his induction strategy) in the proof of Theorem...
Why is factorization of integers important on a first number theory course? Where is factorization used in real life? Are there examples which have a real impact? I am looking for examples which will motivate students.
factor
$\displaystyle 24x^4y^2+28xy^3+30x^5y^2-72xy-6x^5+35x^2y^3-18x^6y-32x^3y+33x^3y^2+63y^3+10x^4y+24x^4-14x^2y$
i have no idea where to start please help me.
Let a positive definite matrix A be factorized to P and Q, A=P*Q and let an arbitrary matrix B.
I am calculating the relative error of the factorization through the norm:
\epsilon = \left\| \textbf{A}-\textbf{PQ} \right\| / \left\| \textbf{A} \right\|
which gives
\epsilon <1\text{e}-16
so I...
Homework Statement
Assume n = p_1*p_2*p_3*...*p_r = q_1*q_2*q_3*...*q_s, where the p's and q's are primes. We can assume that none of the p's are equal to any of the q's. Why?
Homework Equations
The Attempt at a Solution
I am completely stuck on this. My understanding of the...