Complete Factorization

To factorize an expression completely, take the H.C.F outside the bracket and then divide each term with the H.C.F..

Worked Example 5.2.1:

Find the H.C.F of the following and then factorize:

a. 6x and 3y
b. 10xy and  20xy²
c. 12y + 8y² 
d. 36ab  + 12ab²
e. 12p² q – 9pq

Solution

a. 6x and 3y

\( \scriptsize 6x = 2 \: \times \: 3 \: \times \: x \)

\( \scriptsize 3y = 3 \: \times \: y \)

H.C.F  =  3

Take the H.C.F outside the bracket and then divide each term with the HCF

∴ \(\normalsize \frac{6x}{3} \scriptsize =2x \: and \: \normalsize \frac{3y}{3} \scriptsize = y \)

6x and 3y  =  3(2x and y)

b. 10xy and 20xy²

10xy = \( \scriptsize 2 \: \times \: 5 \: \times \: x \: \times \: y \)

20xy²  = \( \scriptsize 2 \: \times \: 2 \: \times \: 5 \: \times \: x \: \times \: y\: \times \: y \)

H.C.F = \( \scriptsize 2 \: \times \: 5 \: \times \: x \: \times \: y = 10xy\)

∴\(\normalsize \frac{10xy}{10xy} \scriptsize =1 \: and \: \normalsize \frac{20xy^2}{10xy} \scriptsize = 2y \)

= 10xy (1 and 2y)

c. 12y + 8y² 

Factors of 12y = 2 × 2 × 3 × y

Factors of 8y² = 2 × 2 × 2 × y × y

H.C.F = 2 × 2 × y = 4y

∴\(\normalsize \frac{12y}{4y} \scriptsize = 3 \: and \: \normalsize \frac{8y^2}{4y} \scriptsize = 2y \)

= 4y (3 + 2y)

d. 36ab + 12ab²

36ab  =  2 × 2 × 3 × 3 × a × b

12ab² = 2 × 2 × 3 × a × b × b

HCF =  2 × 2 × 3 × a × b = 12ab

∴\(\normalsize \frac{36ab}{12ab} \scriptsize = 3 \: and \: \normalsize \frac{12ab^2}{12ab} \scriptsize = b \)

= 12ab (3 + b)

e. 12p² q – 9pq

12p² q = 2 × 2 × 3 × p × p × q

9pq = 3 × 3 × p × q

HCF  =  3 × p × q =  3pq

∴\( \normalsize \frac{12p^2q}{3pq} \scriptsize = 4p \: and \: \normalsize \frac{9pq}{3pq} \scriptsize = 3\)

= 3pq(4p – 3)