Question

How do you factor by grouping $mr+ns-nr-ms$?

Answer 1

Hint: The terms of the given polynomial must first be rearranged so that we can make the groups of two-two terms of the four terms properly. The grouping must be done such that we can take a factor common from each group. Therefore, we can rearrange the terms of the given polynomial as $mr-ms+ns-nr$, so that we can form the two groups by combining the first two and the last two terms as $\left( mr-ms \right)+\left( ns-nr \right)$. Then, we can take the factor of $m$ common from the first group and the factor of $n$ common from the second factor. Then finally, on taking the last common factor outside, the given polynomial will be factored.

Complete step by step solution:
Let us write the given polynomial as
$\Rightarrow p=mr+ns-nr-ms$
By the method of the factor by grouping, we combine the terms of a given polynomial to form the groups which have a factor common to the terms. But we cannot combine the terms in the form in which they are written in the above polynomial. So we rearrange its terms by shifting the last term of $-ms$ to the second position to get
$\Rightarrow p=mr-ms+ns-nr$
Now, we can group the first two and the last two terms to get
$\Rightarrow p=\left( mr-ms \right)+\left( ns-nr \right)$
We can see that the factor of $m$ is common to the first group and the factor of $n$ to the second group. On taking these outside from the respective groups, we get
$\Rightarrow p=m\left( r-s \right)+n\left( s-r \right)$
We can write $s-r=-\left( r-s \right)$ in the second group to get
\[\Rightarrow p=m\left( r-s \right)-n\left( r-s \right)\]
Now, we take the common factor \[\left( r-s \right)\] outside to get
\[\Rightarrow p=\left( r-s \right)\left( m-n \right)\]
Hence, the given polynomial is factored.

Note:
The rearrangement of the terms done by shifting the fourth term to the second position in the above solution is not the only way possible. We can also rearrange the third term $-nr$ to the second position to get $mr-nr+ns-ms$. Then on taking $r$ and $s$ common, we will obtain the factored form of the given polynomial.