Factorization: When an algebraic expression can be written as a product of two or more expressions, then each of these expressions is called a factor of the given expression. And the process is called factorization.

For example:

   (x^2-25)=(x-5)(x+5)  

or

   x^2+9+6x=(x+3)(x+3)  

 

H.C.F of Monomials

H.C.F of Monomials= (H.C.F. of numerical coefficient) × (H.C.F. of Literal coefficient)

Let’s do an example:

   6xy^3 \  and \ 9x^2 y^2= (H.C.F\ of \ 6 \ and \ 9)\times (H.C.F. \ of \ xy^3  \ and\ x^2 y^2 )= 3xy^2  

 

Factorization of an expression by taking out the common factor

Case 1

When the expression is in the form of   ax+by     then proceed as follows:

Step 1: Find the HCF of all the terms of the expression

Step 2: Divide each of the terms with the HCF obtained in step 1

Let’s do an example. Factorize    6x^2-8xy+4x  

Step 1: Find the HCF of    6x^2,  \ 8xy, \ 4x \ which \ is \ 2x 

Step 2: Therefore, 2x  is common in all the terms.

Hence, 6x^2-8xy+4x=2x(6x-4y+2)

Case 2

In case if a polynomial is a common multiplier of each term of the given expression, then first take the common multiplier and then use distributive law.

Expression would look something like this…a(x+y) \pm b(x+y) . In this case (x+y)   is common and we could take that out

Let’s do one example for Case 2. Factorize:

   3a(x+2y)-2b(x+27)=(x+2y)(3a-2b)  

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Factorization of an expression by Grouping the Terms

The expression of the form    ac + bd + ad + bc = a(c + d) + b(c + d) = (a + b)(c + d)    

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Factorizing the difference of two squares

Algebraic expressions like    (a^2-b^2 )=(a+b)(a-b)      

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Factorization of perfect square trinomials

The algebraic expressions of the form

   a^2+b^2 + 2ab \ or \ a^2+b^2-2ab         can be factorize using the formula    (a+b)^2=a^2+b^2+2ab        or    (a-b)^2=a^2+b^2-2ab      

Example:   x^2+14x+49= (x+7)^2     

 

Factorization of Trinomials of the form   Ax^2+Bx+C     

In such a case, find two numbers    a      and   b       such that   a+b=B         and   ab=AC     

Let’s do an example for this as well.

Factorize,     3x^2+11x+10     

Let   a          and   b        be two numbers

  a + b = 11     

  ab = 30     

calculating for   a        and   b        we get   6        and   5     

  3x^2+11x+10     

  =3x^2+6x+5x+10     

  =3x(x+2)+5(x+2)     

  =(3x+5)(x+2)     

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