Factorization means rewriting an expression as a product of factors (things multiplied together).

In simple words:

➡️ You’re turning an expression that’s being added/subtracted into something being multiplied.

⭐ Why Factor?

Factoring helps you:
✅ simplify expressions
✅ solve equations (especially quadratics)
✅ find x-intercepts on graphs
✅ reduce algebraic fractions
✅ understand patterns faster

🔥 The Main Factorization Procedures (Beginner-Friendly)

1️⃣ Greatest Common Factor (GCF)

This is the most common starting method.

Example:

6x + 12

GCF of 6x and 12 is 6

Factor out 6:

6(x+2)

✅ Done.

Example with variables:

8x^2 + 4x

GCF is 4x

4x(2x+1)

2️⃣ Factoring by Grouping

Used when there are 4 terms.

Example:

ax+ay+bx+by

Group:

(ax+ay)+(bx+by)

Factor each group:

a(x+y)+b(x+y)

Now factor out (x+y):

(a+b)(x+y)

3️⃣ Difference of Squares

This is a pattern you should memorize:

a^2 – b^2 = (a-b)(a+b)

Example:

x^2 – 25

Because 25 = 5^2:

(x−5)(x+5)

4️⃣ Trinomial Factoring (x² + bx + c)

This is the classic factoring method.

Example:

x^2 + 7x + 12

Find two numbers that:

• multiply to 12

• add to 7

Those numbers are 3 and 4

So:

(x+3)(x+4)

5️⃣ Trinomial Factoring (ax² + bx + c)

Harder version.

Example:

2x^2 + 7x + 3

Try factors of 2x^2: (2x)(x)
Try factors of 3: (3)(1)

Test:

(2x+1)(x+3)

Multiply to check:

• 2x⋅x=2x^2

• 2x⋅3=6x

• 1⋅x=x

• 1⋅3=3

Combine middle terms: 6x + x = 7✅

So:

2x^2 + 7x + 3 = (2x+1)(x+3)

6️⃣ Perfect Square Trinomials

Another pattern:

a^2 + 2ab + b^2 = (a+b)^2

a^2 – 2ab + b^2 = (a-b)^2

Example:

x^2 + 10x + 25

This is:

(x+5)^2

🧠 Factoring Tip: Always Check for GCF First

Example:

3x^2 + 12x

Factor out GCF (3x):

3x(x+4)

If you skip that step, you might miss the simplest answer.