Question

How do you factor \[{{a}^{2}}+4b-ab-4a\]?

Answer 1

Hint: First of all, we should identify the common parts in different terms. Now we should share these parts from these terms. We should continue this process until the given expression is converted into a collection of linear factors. In this way, we can factor the given expression into linear factors.

Complete step by step solution:
From the question, it is clear that we have to find the factor of \[{{a}^{2}}+4b-ab-4a\]. Let us assume that
\[f(a,b)={{a}^{2}}+4b-ab-4a\]
Let us assume
\[f(a,b)={{a}^{2}}+4b-ab-4a....(1)\]
Now let us rearrange the terms in equation (1). Then we get,
\[\Rightarrow f(a,b)={{a}^{2}}-ab+4b-4a\]
Now we can take as common in the first two terms and 4 as common in other terms. Then, we get the equation as follows
\[\begin{align}
  & \Rightarrow f(a,b)=a(a-b)+4(b-a) \\
 & \Rightarrow f(a,b)=a(a-b)-4(a-b) \\
\end{align}\]
Now let us assume that,
\[\Rightarrow f(a,b)=a(a-b)-4(a-b).....(2)\]
Now we can take \[a-b\] as common in equation (2). Now we get the equation as follows.
\[\Rightarrow f(a,b)=(a-b)(a-4)\]
Now let us assume that,
\[\Rightarrow f(a,b)=(a-b)(a-4).....(3)\]
From equation (3), it is clear that the factors \[{{a}^{2}}+4b-ab-4a\] are \[(a-b)\] and \[(a-4)\].
So, we can write the solution for the given question as follows.
\[\Rightarrow {{a}^{2}}+4b-ab-4a=(a-b)(a-4)\].
Hence, in this way we can factor \[{{a}^{2}}+4b-ab-4a\].

Note:
Students should avoid calculation mistakes while solving this problem and while taking terms as common. Because if a small mistake is made, then the final answer will get interrupted. So, these mistakes should be avoided while solving this problem. Students should also have a clear view of the concept. For example in equation 1 if we take \[-ab\] and \[-4a\]into consideration without rearrangement the solution will be wrong.