Question

How do you complete the factor \[100 - {x^2}{y^2}\]?

Answer 1

Hint: In order to solve this question first, we make a perfect square on both sides of equal to and then use some algebraic identity and then on splitting both the terms we get the factorization of the given mathematical operators.

Complete step-by-step answer:
Given,
\[100 - {x^2}{y^2}\]
To find,
The factors of \[100 - {x^2}{y^2}\].
On making the perfect square on both the side-
100 can be written as \[{10^2}\] and \[{x^2}{y^2}\] can be written as \[{(xy)^2}\]
So on replacing with these we get the expression in form of \[{a^2} - {b^2}\]
\[100 - {x^2}{y^2} = {10^2} - {\left( {xy} \right)^2}\]
Now using the identity \[{a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\]
On comparing the equation and the identity we get the value of a and b then put the value of a and b in the identity.
\[{10^2} - {\left( {xy} \right)^2} = \left( {10 + xy} \right)\left( {10 - xy} \right)\]
Now the obtained answer is the factorization of the given expression.
Final answer:
the factorization of the mathematical expression \[100 - {x^2}{y^2}\] is :
\[ \Rightarrow {10^2} - {\left( {xy} \right)^2} = \left( {10 + xy} \right)\left( {10 - xy} \right)\]

Note: This question is very easy but some students commit mistakes regularly. To solve these types of questions we must know all the algebraic identities and have a knowledge of the use of those identities. Students commit mistakes in factoring the x and y term separately; it means they will not use the combined factor of x and y. Students are also not making the square of both the terms and directly try to split terms and try to factorize. All the algebraic identities include the higher power also. If they don’t know the identities this is very difficult to factorize the given mathematical expression.