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: $3a(x+2y)-2b(x+27)=(x+2y)(3a-2b)$ $\\$

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 $a + b = 11$ $ab = 30$

calculating for $a$  and $b$  we get $6$  and $5$ $3x^2+11x+10$ $=3x^2+6x+5x+10$ $=3x(x+2)+5(x+2)$ $=(3x+5)(x+2)$