Factorization is a fundamental concept in algebra that helps us simplify expressions and solve equations. If you’ve ever struggled with breaking down an algebraic expression, this post is for you!

We’ll start with the essential building block: finding the Highest Common Factor (H.C.F.) of monomials, and then introduce the core idea of Factorization.

💡 What is the H.C.F. (or G.C.D.) of Monomials?

The Highest Common Factor (H.C.F.) of given monomials is the common factor that possesses the greatest coefficient and the highest powers of the variables.

In simpler terms, it’s the biggest expression that divides into all the monomials without leaving a remainder.

🎯 Example 1: Finding H.C.F.

Problem: Find the H.C.F. of $6x^3y$ and $18x^2y^3$.

Solution:

1. Prime Factorize each monomial:

   • $6x^3y = 2 \times 3 \times x \times x \times x \times y$

   • $18x^2y^3 = 2 \times 3 \times 3 \times x \times x \times y \times y \times y$

2. Identify the common factors:

   • Coefficients: The common factors are $2$ and $3$.

   • Variables: The common factors are $x^2$ (two $x$‘s) and $y$ (one $y$).

3. Multiply the common factors:
   $$\text{H.C.F.} = (2 \times 3) \times (x \times x) \times y$$
   $$\text{H.C.F.} = 6x^2y$$

🧠 Quick Check Examples:

Here are a few more to test your understanding:

• H.C.F. of $5xy$ and $10x$ is $5x$

• H.C.F. of $12a^2b$ and $15ab^2$ is $3ab$

• H.C.F. of $2x$ and $4$ is $2$

• H.C.F. of $12x$ and $36$ is $12$

• H.C.F. of $14pq$ and $35pqr$ is $7pq$

✍️ Introduction to Factorization

Now that we know how to find the H.C.F., we can use it for factorization.

Definition: Factorization

The process of writing an algebraic expression as the product of two or more factors is called factorization.

In essence, it’s the reverse process of multiplication (or distribution).

📝 Key Factorization Rules (Notes)

Here are the basic forms of factorization, often achieved by taking out a common factor:

1. Common Factor (Addition):
   $$ab + ac = a(b+c)$$

   (Here, $a$ is the common factor)

2. Common Factor (Subtraction):
   $$ab – ac = a(b-c)$$

   (Again, $a$ is the common factor)

3. Grouping Terms: When you have four terms, sometimes you can group them to find a common binomial factor:
   $$ac + ad + bc + bd$$
   $$= a(c+d) + b(c+d)$$
   $$= (a+b)(c+d)$$
4. Difference of Two Squares: This is one of the most important factorization identities:
   $$a^2 – b^2 = (a+b)(a-b)$$

5. No.	Identity (Standard Form)	Factored Form	Name
   5	$a^2 + b^2 + 2ab$	$(a+b)^2$	Square of a Sum
   6	$a^2 + b^2 – 2ab$	$(a-b)^2$	Square of a Difference
   7	$x^2 + (a+b)x + ab$	$(x+a)(x+b)$	Factoring Trinomials
   8	$a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$	$(a+b+c)^2$	Square of a Trinomial
   9	$a^2 + b^2 + c^2 + 2ab – 2bc – 2ca$	$(a+b-c)^2$	Variation of Square of a Trinomial
   10	$a^3 + b^3 + 3a^2b + 3ab^2$	$(a+b)^3$	Cube of a Sum
   11	$a^3 – b^3 – 3a^2b + 3ab^2$	$(a-b)^3$	Cube of a Difference
   12	$a^3 + b^3$	$(a+b)(a^2 – ab + b^2)$	Sum of Cubes
   13	$a^3 – b^3$	$(a-b)(a^2 + ab + b^2)$	Difference of Cubes
   14	$a^3 + b^3 + c^3 – 3abc$	$(a+b+c)(a^2 + b^2 + c^2 – ab – bc – ca)$	Sum of Cubes (Trinomial)
   15	Conditional Identity: If $a+b+c = 0$	$a^3 + b^3 + c^3 = 3abc$

   🛠️ Examples of Factorization by Common Factor

   The bottom section provides further examples of factoring expressions by extracting the H.C.F. (Highest Common Factor) from the terms.

   1. Simple Common Factor

   Expression: $2x + 4$

   Solution:

   The H.C.F. of $2x$ and $4$ is $2$.

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

   2. Common Factor with Variables

   Expression: $5xy + 10x$

   Solution:

   The H.C.F. of $5xy$ and $10x$ is $5x$.

   $$5xy + 10x = 5x y + 5x (2)$$
   $$= 5x(y + 2)$$

   3. Factorizing with Multi-Term H.C.F.

   Expression: $12a^2b + 15ab^2$

   Solution:

   1. Find H.C.F. of Coefficients (12 and 15): The G.C.D. is $3$.

   2. Find H.C.F. of Variables ($a^2b$ and $ab^2$): The lowest power of $a$ is $a$, and the lowest power of $b$ is $b$. H.C.F. is $ab$.

   3. Overall H.C.F. is $3ab$.

   $$12a^2b + 15ab^2 = (3ab)(4a) + (3ab)(5b)$$
   $$= 3ab(4a + 5b)$$