Question

Find the square by using the identities \[{{\left( xy+3z \right)}^{2}}\]
\[\begin{align}
  & A.{{y}^{2}}x{{y}^{2}}+6xyz+9x{{z}^{2}} \\
 & B.{{x}^{3}}{{y}^{2}}+6z+9{{x}^{2}} \\
 & C.{{x}^{2}}{{x}^{2}}+6xz+9x{{z}^{2}} \\
 & D.{{x}^{2}}{{y}^{2}}+6xyz+9{{z}^{2}} \\
\end{align}\]

Answer 1

Hint: In order to find the square of \[{{\left( xy+3z \right)}^{2}}\] by using identities, firstly we must be checking out the identity that relates to the given polynomial. After checking out the proper identity to be used, we will be expanding the given problem accordingly and we will be obtaining the required answer.

Complete step by step answer:
Now let us learn all about squares. We know that squares of a number are obtained by multiplying the number itself. The number that is being multiplied is called as the square root of the square of that number. The general form of representing a square of the number is \[{{n}^{2}}\]. There exist squares for positive numbers as well as negative numbers. The squares of the positive numbers are generally termed as perfect squares. There also exist imperfect squares. These are generally in the form of decimals.
Now let us find the square of the given polynomial.
We are given with the polynomial \[{{\left( xy+3z \right)}^{2}}\]
Let us check out the identity that \[{{\left( xy+3z \right)}^{2}}\] is related to.
We can see that the identity \[{{\left( a+b \right)}^{2}}\] relates to the polynomial.
The expansion of \[{{\left( a+b \right)}^{2}}\] is \[{{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab\]
Now let us expand \[{{\left( xy+3z \right)}^{2}}\] in the form of \[{{\left( a+b \right)}^{2}}\].
We get,
\[{{\left( xy+3z \right)}^{2}}={{\left( xy \right)}^{2}}+{{\left( 3z \right)}^{2}}+2\left( xy \right)\left( 3z \right)\]
Upon solving this, we get
\[\begin{align}
  & {{\left( xy+3z \right)}^{2}}={{\left( xy \right)}^{2}}+{{\left( 3z \right)}^{2}}+2\left( xy \right)\left( 3z \right) \\
 & \Rightarrow {{x}^{2}}{{y}^{2}}+9{{z}^{2}}+6xyz \\
\end{align}\]
\[\therefore \] The square of \[{{\left( xy+3z \right)}^{2}}\] using identity is \[{{x}^{2}}{{y}^{2}}+9{{z}^{2}}+6xyz\].

So, the correct answer is “Option D”.

Note: In order to expand any polynomial with respect to identity, we must properly check for the identity for an accurate answer. We must note that the nature of a positive number or negative number is always positive in nature.