Complete Factorization

To factorize an expression, divide each term of the expression by its common factor.

To factorize completely, use the Highest Common Factor (HCF). 

Example 3.4.1:

Factorize completely 

(a) 2a + 4ab 
(b) 5x + 10y
(c) 15mn² – 10m²n²
(d) \( \frac{2 \pi r}{T} \: + \: \frac{2 \pi r^2}{T^2} \: – \: \frac{2 \pi r^3}{T^3} \)

Solution 

(a) 2a + 4ab 

Step 1: find the HCF of 2a and 4ab 

 HCF = 2 × a = 2a 

⇒ 2a + 4ab

= 2a (1 + 2b) → last

(b) 5x + 10y 

Find the HCF of 5x and 10y  

HCF = 5

⇒ 5x + 10y

= 5(x + 2y) → last

(c) 15mn³ – 10m²n²

Find the HCF 

HCF = 5 × m × n × n

= 5mn²

⇒ 15mn³ – 10m²n²

= 5mn² (3n – 2m) → last

(d) \( \frac{2 \pi r}{T} \: + \: \frac{2 \pi r^2}{T^2} \: – \: \frac{2 \pi r^3}{T^3} \)

2	\( \frac{2 \pi r}{T} \)	+ \( \frac{2 \pi r^2}{T^2} \)	– \( \frac{2 \pi r^3}{T^3} \)
π	\( \frac{ \pi r}{T} \)	+ \( \frac{ \pi r^2}{T^2} \)	– \( \frac{\pi r^3}{T^3} \)
r	\( \frac{r}{T} \)	+ \( \frac{ r^2}{T^2} \)	– \( \frac{r^3}{T^3} \)
\( \frac{1}{T} \)	\( \frac{1}{T} \)	+ \( \frac{ r}{T^2} \)	– \( \frac{r^2}{T^3} \)
	1	+ \( \frac{ r}{T} \)	– \( \frac{r^2}{T^2} \)

H.C.F = \( \scriptsize 2 \: \times \: \pi \: \times \: r \: \times \: \normalsize \frac{1}{T} \)

H.C.F = \(\frac{2 \pi r}{T} \)

⇒ \( \frac{2 \pi r}{T} \: + \: \frac{2 \pi r^2}{T^2} \: – \: \frac{2 \pi r^3}{T^3} \)

= \(\frac{2 \pi r}{T} \left (\scriptsize 1 \: + \: \normalsize \frac{r}{T} \: -\: \frac{r^2}{T^2} \right)\)

Evaluation Questions: