Question

Factorise $ a{b^2} + (a - 1)b - 1 $
 A) $ (b + 1)(a - 1) $
 B) $ (b + 1)(b - 1) $
 C) $ (b + 1)(ab - 1) $
 D) $ (b - 1)(ab - 1) $

Answer 1

Hint: Factorization of any polynomials can be expressed as the product of its factors having the degree less than or equal to the original polynomial. Here we will expand the bracket and then will find its factors by taking the common terms outside the paired terms for the required solution.

Complete step by step solution:
Take the given expression: $ a{b^2} + (a - 1)b - 1 $
Expand the bracket in between by multiplying the term with the terms inside the bracket with the term outside. When there is a positive sign outside the bracket then the sign of the terms remains the same and there is no change in the sign of the terms.
 $ = a{b^2} + ab - b - 1 $
Make the pair of first two terms and the last two terms.
 $ = \underline {a{b^2} + ab} - \underline {b - 1} $
Find the common multiple from the paired terms in the above expression –
 $ = ab(b + 1) - 1(b + 1) $
Take the common factor common from the above expression –
 $ = (b + 1)(ab - 1) $
From the given multiple choices, the option C is the correct answer.
So, the correct answer is “ (b + 1)(ab - 1)”.

Note: Be careful about the sign convention while expansion of the brackets. When there is a negative sign outside the bracket then the sign of the terms inside the bracket changes. Positive terms become negative and negative terms become positive. While when there is positive sign outside the bracket then there is no change in the sign of the terms inside the bracket when opened. Here we already had four terms to make a pair of two – two terms, in case if you have only three terms then split the middle term and find its factors.