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Factorization
  
Introduction
  

    Factorization is the process of breaking down an algebraic expression or a number into a product of simpler factors. It is a fundamental concept in algebra and is widely used in solving equations, simplifying expressions, and finding roots of polynomials.
  

  
Basic Concepts
  

    
Factor: A factor is a number or expression that divides another number or expression completely without leaving a remainder.
    
Prime Factorization: Breaking down a number into a product of prime numbers.
    
Factorization of Polynomials: Breaking down a polynomial into a product of simpler polynomials.
  

  
  
1. Common Factor Method
  
Example 1: Factorize 
    $12x{y}^2-18x{y}^3$
  
  
Solution:
  
GCF of $12$ and $18$ is $6$, and common variables are $x$ and $y^2$.
  
$12x{y}^2-18x{y}^3=6x{y}^2(2-3y)$

  
Example 2: Factorize 
    $5a(x+y)+10b(x+y)$
  
  
Solution:
  
$=5(x+y)(a+2b)$

  
  
2. Grouping Method
  
Example 1: Factorize 
    $x^3+x^2+x+1$
  
  
Solution:
  
Group terms: $(x^3+x^2)+(x+1)$
  
Factor: $x^2(x+1)+1(x+1)=(x+1)(x^2+1)$

  
Example 2: Factorize $2ab-3ac+4bd-6cd$
  
Solution:
  

    Regroup: 
    $(2ab-3ac)+(4bd-6cd)$
  
  

    Factor: $a(2b-3c)+2d(2b-3c)=(2b-3c)(a+2d)$
  

  
  
3. Difference of Squares
  
Example 1: Factorize $25a^2-49b^2$
  
Solution:
  
$={\left(5a\right)}^2-{\left(7b\right)}^2=(5a-7b)(5a+7b)$

  
Example 2: Factorize $x^4-16$
  
Solution:
  
$={\left(x^2\right)}^2-4^2=(x^2-4)(x^2+4)$
  
Further factorize $x^2-4$: $(x-2)(x+2)(x^2+4)$.

  
  
4. Perfect Square Trinomials
  
Example 1: Factorize 
    $4x{y}^2+12xy+9x$
  
  
Solution:
  

    Factor out 
    $x$: 
    $x(4y^2+12y+9)$
  
  

    Factor the trinomial: 
    $4y^2+12y+9={(2y+3)}^2$
  
  

    Final result: 
    $x\left({(2y+3)}^2\right)$.
  

  
Example 2: Factorize 
    $x^2-10x+25$
  
  
Solution:
  

    $={(x-5)}^2$
  

  
  
5. Quadratic Trinomials
  
Example 1: Factorize 
    $2x{y}^2+7xy+3x$
  
  
Solution:
  
Factor out $x$: $x(2y^2+7y+3)$
  
Find factors of $2\times 3=6$ that add to $7$: $6$ and $1$.
  

    Rewrite: 
    $2y^2+6y+y+3$
  
  

    Factor: 
    $2y(y+3)+1(y+3)=(y+3)(2y+1)$
  
  

    Final result: 
    $x(y+3)(2y+1)$
  

  
Example 2: Factorize 
    $x^2-3x-10$
  
  
Solution:
  
Find two numbers that multiply to $\mathrm{-10}$ and add to $\mathrm{-3}$: $\mathrm{-5}$ and $2$.
  

    Rewrite: 
    $x^2-5x+2x-10$
  
  

    Factor: 
    $x(x-5)+2(x-5)=(x-5)(x+2)$
  

  
  
Mixed Problems
  

    

      
Factorize 
        $4x{y}^2-9x$
      
      
Solution:
      

        Factor out 
        $x$: 
        $x(4y^2-9)$
      
      

        Apply difference of squares: 
        $x(2y-3)(2y+3)$
      
    
    

      
Factorize 
        $x^3-8$
        (Difference of Cubes)
      
      
Solution:
      

        Use formula: 
        $a^3-b^3=(a-b)(a^2+\mathrm{ab}+b^2)$
      
      

        $=(x-2)(x^2+2x+4)$
      
    
    

      
Factorize 
        $a^3-b^3$
        (Difference of Cubes)
      
      

        Use formula: 
        $a^3-b^3=(a-b)(a^2+\mathrm{ab}+b^2)$
      
      

        Example: 
        $m^3-n^3$
      
      

        Solution: 
        $=(m-n)(m^2+\mathrm{mn}+n^2)$
      
    
  

  
Conclusion
  

    Factorization is a crucial skill in algebra and is helpful in various fields of mathematics. The methods outlined here can be applied to many types of expressions and equations, making them easier to simplify or solve. Mastering these techniques provides a solid foundation for tackling more advanced algebraic problems.