Question

Find the factor of the given equation:
$\left( {{x^2} - 2xy + {y^2}} \right) - {z^2}$

Answer 1

Hint:To find the factor of this type of question we have to choose a combination of where we can make a square of some terms and where we can apply some formulae by simple mathematics to make it in factor form.

Complete step-by-step solution:
We have given:
$\left( {{x^2} - 2xy + {y^2}} \right) - {z^2}$
Now we have to convert it in factor form
So we will write it as
$
   \Rightarrow \left( {{x^2} - xy - xy + {y^2}} \right) - {z^2} \\
   \Rightarrow \left[ {x\left( {x - y} \right) - y\left( {x - y} \right)} \right] - {z^2} \\
   \Rightarrow \left( {x - y} \right)\left( {x - y} \right) - {z^2} \\
   \Rightarrow {\left( {x - y} \right)^2} - {z^2} \\
 $
Now we will use the formulae $\left( {{a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)} \right)$
$
   \Rightarrow \left[ {\left( {x - y} \right) + z} \right]\left[ {\left( {x - y} \right) - z} \right] \\
   \Rightarrow \left[ {x - y + z} \right]\left[ {x - y - z} \right] \\
 $
Now this is the factor form of the given question.

Note: Whenever we get this type of question the key concept of solving is we have to use our virtue that what should we do that it becomes simple either by taking common or by adding something or subtracting something do anything just we have to make it in factor form.