Factorisation (copy) (copy) (copy) (copy)

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, slskafss ~~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 {\displaystyle x} can be trivially written as {\displaystyle (xy)\times (1/y)} whenever {\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.

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Factorization

Factorization is the process of finding the factors of a given number. A factor is a number that divides another number without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.

Methods of Factorization

There are several methods of factorization, including:

Trial Division

Trial division is the simplest method of factorization. It involves dividing the number by each integer from 2 to the square root of the number. If any of these divisions result in an integer quotient, then the divisor is a factor of the number.

Prime Factorization

Prime factorization is the process of expressing a number as a product of its prime factors. To find the prime factors of a number, we can use trial division to find the smallest prime factor, then divide the number by that factor and repeat the process until we are left with only prime factors.

Factoring by Grouping

Factoring by grouping is a method used to factor polynomials with four or more terms. It involves grouping terms with common factors and factoring out those common factors.

Factoring by Completing the Square

Factoring by completing the square is a method used to factor quadratic equations. It involves adding and subtracting a constant term to the equation to create a perfect square trinomial, which can then be factored.

Importance of Factorization

Factorization is an important concept in mathematics and has many applications in fields such as cryptography, number theory, and algebra. It is also used in simplifying expressions and solving equations.

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Yes, I am chatGPT. Here is a Python script for finding the missing element from an array:

def find_missing(arr1, arr2):
    """
    This function takes two arrays as input and returns the missing element from the second array.
    """
    # Calculate the sum of all elements in arr1
    sum_arr1 = sum(arr1)

    # Calculate the sum of all elements in arr2
    sum_arr2 = sum(arr2)

    # Return the difference between the two sums
    return sum_arr1 - sum_arr2

# Example usage
arr1 = [1, 2, 3, 4, 5, 6]
arr2 = [1, 2, 3, 4, 6]
print(find_missing(arr1, arr2)) # Output: 5

The main theme of this script is to find the missing element from an array. The script achieves this by calculating the sum of all elements in both arrays and returning the difference between the two sums. The script assumes that one element is missing from the second array and that the two arrays have the same elements except for the missing one.

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Factorisation

• Central Idea: Breaking down a number into its factors

Methods

• Main Branch: Different methods for factorisation

Factor Tree

• Sub-branch: Creating a factor tree to find prime factors
  • Example: 48
  • 2
    • 2
    • 2
    • 3
  • 2 x 2 x 2 x 3 = 48

Division Method

• Sub-branch: Dividing the number by its factors to find prime factors
  • Example: 72
  • 72 ÷ 2 = 36
    • 36 ÷ 2 = 18
    • 18 ÷ 2 = 9
      • 9 ÷ 3 = 3
      • 3 ÷ 3 = 1
  • 2 x 2 x 2 x 3 x 3 = 72

Common Factor Method

• Sub-branch: Finding common factors between two or more numbers
  • Example: 24 and 36
  • Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
  • Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
  • Common factors: 1, 2, 3, 4, 6, 12
  • Highest common factor: 12

Applications

• Main Branch: Applications of factorisation

Simplifying Fractions

• Sub-branch: Using factorisation to simplify fractions
  • Example: 24/36
  • Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
  • Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
  • Common factors: 1, 2, 3, 4, 6, 12
  • Divide numerator and denominator by highest common factor (12)
  • Simplified fraction: 2/3

Solving Equations

• Sub-branch: Using factorisation to solve equations
  • Example: x

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