Factorization

Factorization

Hello ScienceBee, let us talk about factorization in this lesson. 
 

 

What is factorization?   

Factorization (also factorisation in some forms of British English) 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.
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.
Video: Factorization

 

How to factor integers? 

By the fundamental theorem of arithmetic, every integer greater than 1 has a unique (up to the order of the factors) factorization into prime numbers, which are those integers which cannot be further factorized into the product of integers greater than one.
For computing the factorization of an integer n, one needs an algorithm for finding a divisor q of n or deciding that n is prime. When such a divisor is found, the repeated application of this algorithm to the factors q and n / q gives eventually the complete factorization of n.

 

Factoring example. 

For factoring n = 1386 into primes:
  • Start with division by 2: the number is even, and n = 2 · 693. Continue with 693, and 2 as a first divisor candidate.
  • 693 is odd (2 is not a divisor), but is a multiple of 3: one has 693 = 3 · 231 and n = 2 · 3 · 231. Continue with 231, and 3 as a first divisor candidate.
  • 231 is also a multiple of 3: one has 231 = 3 · 77, and thus n = 2 · 32 · 77. Continue with 77, and 3 as a first divisor candidate.
  • 77 is not a multiple of 3, since the sum of its digits is 14, not a multiple of 3. It is also not a multiple of 5 because its last digit is 7. The next odd divisor to be tested is 7. One has 77 = 7 · 11, and thus n = 2 · 32 · 7 · 11. This shows that 7 is prime (easy to test directly). Continue with 11, and 7 as a first divisor candidate.
  • As 72 > 11, one has finished. Thus 11 is prime, and the prime factorization is
1386 = 2 · 32 · 7 · 11.

 

Find the prime factorization for 75. 

This video shows how to find the prime factorization for 75.:

 

Find the prime factorization for 36 and 73. 

 This video shows how to find the prime factorization for 36 and 73. 

 

Finding the factors for perfect squares. 

 This video shows how to find the factors for perfect squares. 

 

How to factor sum of squares? 

This video shows how to factor sum of squares.

 

How to factor quadratics? 

 This video shows how to factor quadratics.

  

How to do monomial factorization? 

This video shows how to do monomial factorization.

 

How to factor a polynomial? 

This video shows how to factor a polynomial.

 

How to factor using the ladder method? 

This video shows how to factor using the ladder method.

 

How to factor trinomials?  

This video shows how to factor trinomials.

 

Let's Review

  1. ____ consists of writing a number or another mathematical object as a product of several ____ , usually smaller or simpler objects of the same kind.
  2. 3 × 5 is a factorization of the integer ____ , and (x – 2)(x + 2) is a factorization of the polynomial ____ .
  3. A meaningful factorization for a rational number or a rational function can be obtained by writing it in lowest terms and separately factoring its ____ and ____ .

 

Answer

  1. 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.
  2. 3 × 5 is a factorization of the integer 15, and (x – 2)(x + 2) is a factorization of the polynomial x2 – 4.
  3. 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.

 

Test Your Knowledge

 

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