In mathematics, factor is one of the important topics in algebra. Factor is the process of performing multiplication in reverse. Distributive law is the process of the product of a terms and a sum is equal to the sum of the individual products of the addends and the terms. Let us solve some example problems for factor using the distributive law.

Distributive property:

a . (b + c) = ab + ac

Different steps to solve the factor using distributive law are

- Given expression
- For each term in the expression can determine the greatest common factor.
- Each term can be divided by the greatest common factor
- Product of the greatest common factor and the remaining factor of the each term can be written as distributive law.

- Solution to the given expression

**Example 1:**

Factor the given expression

18a^{2}b^{5} + 6ab^{2} + 24ab

**Solution:**

Given expression

18a^{2}b^{5} + 6ab^{2} + 24ab

Factor the each term

First term in the expression can be factored

18a^{2}b^{5} = 2 . 3 . 3 . a . a . b . b . b . b . b

Second term in the expression can be factored

6ab^{2} = 2 . 3 . a . b . b

Last term in the expression can be factored

24ab = 2 . 2 . 2 . 3 . a . b

Determine the greatest common factor for the expression 6ab

Using the distributive law the expression can be written as

18a^{2}b^{5} + 6ab^{2} + 24ab = 6ab (3ab^{4}) + 6ab (b) + 6ab (4)

= 6ab (3ab^{4} + b + 4)

**Answer:**

18a^{2}b^{5} + 6ab^{2} + 24ab = 6ab (3ab^{4} + b + 4)

**Example 2:**

Factor the given expression

20a^{3}b^{4} + 25a^{2}b^{2} + 10a^{5}b^{3}

**Solution:**

Given expression

20a^{3}b^{4} + 25a^{2}b^{2} + 10a^{5}b^{3}

Factor the each term

First term in the expression can be factored

20a^{3}b^{4} = 2 . 2. 5 . a . a . a . b . b . b . b

Second term in the expression can be factored

25a^{2}b^{2} = 5 . 5 . a . a . b . b

Last term in the expression can be factored

10a^{5}b^{3} = 2 . 5 . a . a . a . a . a . b . b . b

Determine the greatest common factor for the expression 5a^{2}b^{2}

Using the distributive law the expression can be written as

20a^{3}b^{4} + 25a^{2}b^{2} + 10a^{5}b^{3} = 5a^{2}b^{2} (4ab^{2}) + 5a^{2}b^{2} (5) +
5a^{2}b^{2} (2a^{3}b)

= 5a^{2}b^{2} (4ab^{2} + 5 + 2a^{3}b)

**Answer:**

20a^{3}b^{4} + 25a^{2}b^{2} + 10a^{5}b^{3} = 5a^{2}b^{2} (4ab^{2} + 2a^{3}b + 5)