Question

The factors of ${a^2} + b - ab - a$ are-
A.$\left( {a - 1} \right)\left( {a - b} \right)$
B.\[\left( {a + b} \right)\left( {a - 1} \right)\]
C.$\left( {a + 1} \right)\left( {a - b} \right)$
D.None of these

Answer 1

Hint: First observe the function and take those terms together which have something common in them. Then rearrange the function so that such terms come together. Then take the common term out of the first and second term and third and fourth term. Then again take the term formed in brackets as you will get the factors of the function.

Complete step-by-step answer:
Given function is- ${a^2} + b - ab - a$
Now we have to find its factors.
So we will first observe the function and take those terms together which have something common in them. Here we see that ‘a’ is common in both ${a^2}$ first term and fourth term and ‘b’ is common in the second term and third term.
So we can rearrange the function and write-
$ \Rightarrow {a^2} - a - ab + b$
Now on taking ‘a’ common from first and second term and taking ‘b’ common from third and fourth term, we get,
$ \Rightarrow a\left( {a - 1} \right) - b\left( {a - 1} \right)$
Here when we take negative sign common with b, the sign inside the bracket changes from + to – because $\left( - \right) \times \left( - \right) = + $
Now we see that $\left( {a - 1} \right)$ is common in both terms in the above function, so we'll take it common and we will get-
$ \Rightarrow \left( {a - 1} \right)\left( {a - b} \right)$
These are the factors of the given equation.
Hence the correct answer is A.

Note: You can also solve this question by taking first and third term together and second and fourth term together-
$ \Rightarrow {a^2} - ab - a + b$ On taking the common terms from first and second term and third and fourth term we get,
$ \Rightarrow a\left( {a - b} \right) - 1\left( {a - b} \right)$
Here we will take $\left( {a - b} \right)$ common and we will get,
$ \Rightarrow \left( {a - 1} \right)\left( {a - b} \right)$
Hence you can solve the question by any of these methods.