Question

Factorize: \[6ab - {b^2} + 12ac - 2bc\]

Answer 1

Hint:
When we factorize an algebraic expression, we write it as a product of factors. These factors may be number algebraic, variables, or algebraic expressions. An expression like $ 3xy,5{x^2}y $ is already in the factored form. Their factors can be just read off from them, as we already know.
On the other hand, consider expressions like $ 2x + 4, 3x + 3. $ It is not obvious what their factors are.
Factorization by regrouping terms: When the expression will not be easy to see the factorization. Rearranging the expression allows us to form groups to factorizations. This is regrouping.

Complete step by step solution:
Step1: We have Given equation $ $ \[6ab - {b^2} + 12ac - 2bc\]. The given expression is not in factor form, firstly, we have to do rearranging the terms. In the given expression, there are four terms.
Step2: Check, if there is a common factor among all terms. There is none.
Think of grouping; Notice the first two terms, in the first term $ 6ab $ we can write $ 6 \times a \times b $ . In the second term $ {b^2} $ , we can think $ b \times b $ . So, notice that in the first two terms they have common factor b.
We can write this as $ 6 \times a \times b - b \times b = b(6a - b)........(i) $
Notice the last two terms, in the third term $ 12ac $ we can write $ 2 \times 6 \times a \times c $ .
In the fourth term $ - 2bc, $ we can write - $ 2 \times b \times c $ .
So notice that the common factor in the third and fourth term is $ 2 \times c $ .
We can write this as $ 2 \times 6 \times a \times c - 2 \times b \times c = 2b(6a - b).................(ii) $
Step 3: Putting a and b together;
\[6ab - {b^2} + 12ac - 2bc\] $ = b(6a - b) + 2c(6a - b) $
It is converted into two terms. Again, notice that there is a common factor in two terms. The factor is $ (6a - b) $ . So, we can take $ (6a - b) $ as common and the equation becomes

$ (6a - b)(b + 2c) $
This is the required answer.

Note:
We know that $ {(a + b)^2} = {a^2} + 2ab + {b^2} $
 $ {(a - b)^2} = {a^2} - 2ab + {b^2} $ and
 $ (a + b)(a - b) = {a^2} - {b^2} $
Example: $ {x^2} + 8x + 16,4{q^2}p - 36 $ are the examples, how to use these identities of factorization. What we do is to observe the given expression. If it has a form that fits the right-hand side of one of the identities then the expression corresponding to the left-hand side of the identity gives the desired factorization.