Question

How do you factor by grouping $xy+x-y-1$?

Answer 1

Hint: The given polynomial $xy+x-y-1$ consists of a total of four terms. So we can form two pairs of terms from the given four terms by grouping two-two terms together. The first pair can be formed from the first two terms and the second pair can be formed from the last two terms as $\left( xy+x \right)+\left( -y-1 \right)$. Then from these two pairs, we need to take outside the factors $x$ and $-1$ which are common to the respective pairs. In doing so, we will be left with one factor common to each of the two pairs, which on taking outside will completely factor the given polynomial.

Complete step by step solution:
Let us consider the polynomial given in the above question as
$\Rightarrow p\left( x,y \right)=xy+x-y-1$
To use the factor by grouping method, we need to group two-two terms of the four terms together so as to form two pairs of the terms. Therefore we pair up the first two and the last two terms to get
$\Rightarrow p\left( x,y \right)=\left( xy+x \right)+\left( -y-1 \right)$
Now, since the factors $x$ and $-1$ are common to the first and the second pairs, we can take them outside from the respective pairs to get
$\Rightarrow p\left( x,y \right)=x\left( y+1 \right)-1\left( y+1 \right)$
Now, the factor of $\left( y+1 \right)$ can be taken common from the two pairs to get
$\Rightarrow p\left( x,y \right)=\left( y+1 \right)\left( x-1 \right)$
Hence, the given polynomial is completely factored using the factor by grouping method.

Note:
There is no compulsion that we have to form the two pairs by groping the first two and the last two terms only. We can combine any of the two terms in the given polynomial $xy+x-y-1$ so as to factor it by grouping. For example, we can also form pairs by grouping the first term with the third term and the second term with the last term to get $\left( xy-y \right)+\left( x-1 \right)$. Then by proceeding similarly as in the above solution, we will be able to factor it.