Question

How do you factor $4ab+1-2a-2b?$

Answer 1

Hint: We will factor the given expression by grouping. In the first step we regroup the terms in the expression. We will take the common factors out. And rearrange, if necessary.

Complete step by step solution:
Consider the given expression $4ab+1-2a-2b.$
We are going to rearrange the terms of the given expression.
We regroup the terms so that this expression can be reduced into the product of its factors.
First, let us consider the regrouping,
$\Rightarrow 4ab-2b+1-2a.$
That means, the groups after separating the expression with parentheses,
$\Rightarrow \left( 4ab-2b \right)+\left( 1-2a \right).$
From these groups, we are going to factor out the highest common factors or the greatest common divisors. In the first group, the highest common factor is $2b$ and in the second group, the highest common factor is $1.$
Let us factor out the highest common factors, we get,
$\Rightarrow 2b\left( 2a-1 \right)+\left( 1-2a \right).$
Now, we are supposed to take $-1$ out of the first summand. We get,$\Rightarrow -2b\left( 1-2a \right)+\left( 1-2a \right).$
This stage of factoring contains another highest common factor in both the summands. That is, $1-2a.$
We are factoring that out, we get,
$\Rightarrow \left( -2b+1 \right)\left( 1-2a \right).$
A small rearrangement will give us,
$\Rightarrow \left( 1-2b \right)\left( 1-2a \right).$
Also, we can write,
$\Rightarrow \left( 1-2a \right)\left( 1-2b \right).$
Therefore, the factors of the given expression $4ab+1-2a-2b$ are $1-2a$ and $1-2b.$
Hence the factorization is given as, $4ab+1-2a-2b=\left( 1-2a \right)\left( 1-2b \right).$

Note: Given below is an alternative method to regroup and factor the given expression:
The given expression is \[4ab+1-2a-2b.\]
The regrouping is done as,
$\Rightarrow 4ab-2a+1-2b.$
Parenthesised the groups as follows,
$\Rightarrow \left( 4ab-2a \right)+\left( 1-2b \right).$
Let us factor out the greatest common divisors from these groups. In the first group, the highest common factor is $-2a$ and in the second group, the highest common factor is $1.$
We get,
$\Rightarrow -2a\left( 1-2b \right)+\left( 1-2b \right).$
There is a greatest common factor in both the groups of above obtained expression, $1-2b.$
We factoring out the GCD,
$\Rightarrow \left( -2a+1 \right)\left( 1-2b \right).$
We can rearrange this as,
$\Rightarrow \left( 1-2a \right)\left( 1-2b \right).$
Therefore, the factors of the given expression $4ab+1-2a-2b$ are $1-2a$ and $1-2b.$
Hence the factorization is given as, $4ab+1-2a-2b=\left( 1-2a \right)\left( 1-2b \right).$