Question

Factorise the given equation: \[6ab - {b^2} + 12ac - 2bc\]

Answer 1

Hint: Here, we will use the concept of the factorisation. Factorisation is the process in which a number is written in the forms of its small factors which on multiplication give the original number. Here we will take the common terms from the first two terms of the given equation. Then we will factor out a common term from the last two terms of the equation. By this way we will get the factors of the given equation.

Complete step by step solution:
The given equation is \[6ab - {b^2} + 12ac - 2bc\].
Firstly, we will take \[b\] common from the first two terms of the equation because that’s the common term in the first two terms of the equation. Therefore, we get
\[ \Rightarrow 6ab - {b^2} + 12ac - 2bc = b\left( {6a - b} \right) + 12ac - 2bc\]
Now we will take \[2c\] common from the last two terms of the given equation. Therefore, we get
\[ \Rightarrow 6ab - {b^2} + 12ac - 2bc = b\left( {6a - b} \right) + 2c\left( {6a - b} \right)\]
Now we can clearly see that the term \[\left( {6a - b} \right)\] is common in both the terms. Therefore taking common \[\left( {6a - b} \right)\], we get
\[ \Rightarrow 6ab - {b^2} + 12ac - 2bc = \left( {6a - b} \right)\left( {b + 2c} \right)\]
Hence, \[\left( {6a - b} \right),\left( {b + 2c} \right)\] are the factors of the given equation.

Note: Here we will note that we have to take the maximum common terms from the equation and we have to simplify the equation carefully to get the factors. Factors can be the same but in our case it’s different. We should know that the factors we have obtained on solving that we will get the original given equation. Factors are the smallest part of the number or equation which on multiplication will give us the actual number of equations. Generally in these types of questions algebraic identities are used to solve and make the factors of the equation. Algebraic identities are equations where the value of the left-hand side of the equation is identically equal to the value of the right-hand side of the equation. It is very important that we learn about algebraic identities in maths.