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.


   (x^2-25)=(x-5)(x+5)      or


H.C.F of Monomials

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


   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.


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:


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)    

Factorizing the difference of two squares

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

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

Therefore   a + b = 11    and     ab = 30 . Hence calculating for   a and   b we get   6 and   5 . Therefore