Question

Factorize the expression $\left( {1 - {x^2}} \right)\left( {1 - {y^2}} \right) + 4xy$.

Answer 1

Hint:Group all the like terms and gather them at one place. Split the terms in order to frame the equation into any kind of mathematical formula. Then after shortening them using formulae.

Complete step-by-step answer:
We are asked to find out the factors.
Initially, it's better to expand them all if we can.
$
  \left( {1 - {x^2}} \right)\left( {1 - {y^2}} \right) + 4xy = 1 - {y^2} - {x^2} + {x^2}{y^2} + 4xy \\
    \\
 $
Now, if you look at the equation carefully, if we split the term $4xy$ into $2xy + 2xy$ we will get one formula as follows
$
   \Rightarrow 1 - {y^2} - {x^2} + {x^2}{y^2} + 2xy + 2xy \\
    \\
 $
Rearrange the term to easily visualize the formula in the equation
$
   \Rightarrow 1 + {x^2}{y^2} + 2xy - {y^2} - {x^2} + 2xy \\
    \\
 $
Here you can see the term which looks like ${a^2} + {b^2} + 2ab$ where $a$ is $1$ and $b$ is $xy$
As we know ${a^2} + {b^2} + 2ab = {\left( {a + b} \right)^2}$ implement them with the above values and take negative common from remaining terms, we get
$
   \Rightarrow {\left( {1 + xy} \right)^2} - \left( {{y^2} + {x^2} - 2xy} \right) \\
    \\
 $
Now, in the second part also you can apply the same formula with $a$ as $x$ and $b$ as $ - y$ them we get
$
   \Rightarrow {\left( {1 + xy} \right)^2} - {\left( {x - y} \right)^2} \\
    \\
 $
Now apply another formula ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$
With $a$ as $1 + xy$ and $b$ as $x - y$
$ \Rightarrow \left( {1 + xy + y - x} \right)\left( {1 + xy - y + x} \right)$
Hence the factorization of $\left( {1 - {x^2}} \right)\left( {1 - {y^2}} \right) + 4xy$ is $\left( {1 + xy + y - x} \right)\left( {1 + xy - y + x} \right)$.

Note:The easy way to tackle this kind of problems is to expand them firstly and look carefully to find out how terms and knowledge to apply different algebraic formulas is also important. Sometimes it’s better to cross-check with some of the predicted factors also.