1. Factorization by Common Factors

In this method, we look for a term that divides all the parts of the expression.

Example:
Factorise ( 8x + 12 )

Solution:
The common factor of 8 and 12 is 4.
So,
( 8x + 12 = 4(2x + 3) )

Further Examples

1. Factorize: ( 6x + 9 )

Step 1: Identify the common factor of 6 and 9.
Common factor = 3

Step 2: Factor out 3:
6x + 9 = 3(2x + 3)

Final Answer:
3(2x + 3)

2. Factorize: ( 12y^2 - 15y )

Step 1: Identify the common factor of 12 and 15.
Common factor = 3y

Step 2: Factor out 3y:
12y^2 - 15y = 3y(4y - 5)

Final Answer:
{3y(4y - 5)}

3. Factorize: ( \frac{5}{2}x - \frac{15}{4} )

Step 1: Find the common factor of the coefficients.
\frac{5}{2} \quad \text{and} \quad \frac{15}{4}
Common factor = (\frac{5}{2})

Step 2: Factor out (\frac{5}{2}):
\frac{5}{2}x - \frac{15}{4}
= \frac{5}{2}\left(x - \frac{15/4}{5/2}\right)

Simplify the inner fraction:
\frac{15/4}{5/2} = \frac{15}{4} \times \frac{2}{5} = \frac{30}{20} = \frac{3}{2}

So,
\frac{5}{2}x - \frac{15}{4} = \frac{5}{2}\left(x - \frac{3}{2}\right)

Final Answer:
{\frac{5}{2}\left(x - \frac{3}{2}\right)}

4. Factorize: ( \frac{8}{9}a^2 + \frac{4}{3}a )

Step 1: Identify the common factor.
\frac{8}{9}a^2 \quad \text{and} \quad \frac{4}{3}a

Common factor of the coefficients:

• (\frac{8}{9}) and (\frac{4}{3}) → common factor = (\frac{4}{9})
  Common algebraic factor = a

So common factor =
\frac{4}{9}a

Step 2: Factor out (\frac{4}{9}a):
\frac{8}{9}a^2 + \frac{4}{3}a
= \frac{4}{9}a\left( \frac{8/9}{4/9}a + \frac{4/3}{4/9} \right)

Simplify inside:

• (\frac{8/9}{4/9} = 2)
• (\frac{4/3}{4/9} = \frac{4}{3} \times \frac{9}{4} = 3)

So,
\frac{8}{9}a^2 + \frac{4}{3}a = \frac{4}{9}a(2a + 3)

Final Answer:
{\frac{4}{9}a(2a + 3)}

5. Factorize: ( 7x - \frac{21}{5} )

Step 1: Identify the common factor.
Common factor of 7 and (\frac{21}{5}) = 7

Step 2: Factor out 7:
7x - \frac{21}{5} = 7\left(x - \frac{21/5}{7}\right)

Simplify inside:
\frac{21/5}{7} = \frac{21}{5} \times \frac{1}{7} = \frac{3}{5}

Thus,
7x - \frac{21}{5} = 7\left(x - \frac{3}{5}\right)

Final Answer:
{7\left(x - \frac{3}{5}\right)}