Question

How do you factor $2x-2y-{{x}^{2}}+2xy-{{y}^{2}}$ ?

Answer 1

Hint: For factoring this polynomial first we need to take the constant common out of the first two terms and then on closely observing we can see that the last three terms are in the form of ${{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab$ . Use this postulate and then further evaluate or group the factors together and write them in the product of sums form. Now write all of them together and represent them as the factors of the given expression.

Complete step by step solution:
The given polynomial is $2x-2y-{{x}^{2}}+2xy-{{y}^{2}}$
It is a polynomial in two variables $x,y\;$
Now let us first take the common terms from the first two terms.
As we can look through, we see that the constant $2$ is common in both terms.
After taking the common factor out we get,
$\Rightarrow 2\left( x-y \right)-{{x}^{2}}+2xy-{{y}^{2}}$
Now let us consider the last three terms.
On closely observing we can see that the last three terms are in the form of ${{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab$
Here $a=x;b=y$
Now let us substitute the formula.
We get,
$\Rightarrow 2\left( x-y \right)-\left( {{x}^{2}}-2xy+{{y}^{2}} \right)$
$\Rightarrow 2\left( x-y \right)-{{\left( x-y \right)}^{2}}$
Now let us consider the term $x-y$
It is common in both terms.
So, it is also one of the factors. On taking the common factor out we get,
$\Rightarrow \left( x-y \right)\left[ 2-\left( x-y \right) \right]$
Further represented as,
$\Rightarrow \left( x-y \right)\left( 2-x+y \right)$
Now we can see that there is no other polynomial left that can be factored further.
Now writing it all together we get,
$\Rightarrow \left( x-y \right)\left( 2-x+y \right)$

Hence the factors for the polynomial $2x-2y-{{x}^{2}}+2xy-{{y}^{2}}$ are $\left( x-y \right)\left( 2-x+y \right)$

Note: The most common mistake students tend to make is rearranging the terms in descending order of the exponentials in the first step itself. One must analyze the question and check if there is any way to use the identities in simplifying the expression and then factoring.