Question

Find the square of the following polynomial $\left( 5{{a}^{2}}+6{{b}^{2}} \right)$.

Answer 1

Hint: Square of the given polynomial is given by ${{\left( 5{{a}^{2}}+6{{b}^{2}} \right)}^{2}}=\left( 5{{a}^{2}}+6{{b}^{2}} \right)\left( 5{{a}^{2}}+6{{b}^{2}} \right)$, we can open the brackets and multiply the terms. Then solve them to get the square of the given polynomial.

Complete step-by-step answer:

In the question we are given the polynomial, $\left( 5{{a}^{2}}+6{{b}^{2}} \right)$
Polynomial is an expression having variables and corresponding coefficients, in the given polynomial, ‘a, b’ are the variables. The coefficient of ‘a’ is 5 and that of ‘b’ is 6. The given polynomial is of second degree, as the power of ‘a’ and ‘b’ is 2.
Here we need to find the square of this given polynomial. So, the square of the given polynomial can be written as,
${{\left( 5{{a}^{2}}+6{{b}^{2}} \right)}^{2}}=\left( 5{{a}^{2}}+6{{b}^{2}} \right)\left( 5{{a}^{2}}+6{{b}^{2}} \right)$
Now opening the brackets, the above equation can be written as,
\[{{\left( 5{{a}^{2}}+6{{b}^{2}} \right)}^{2}}=5{{a}^{2}}\left( 5{{a}^{2}}+6{{b}^{2}} \right)+6{{b}^{2}}\left( 5{{a}^{2}}+6{{b}^{2}} \right)\]
On further simplification we get,
\[{{\left( 5{{a}^{2}}+6{{b}^{2}} \right)}^{2}}=25{{a}^{4}}+(5{{a}^{2}})(6{{b}^{2}})+(5{{a}^{2}})(6{{b}^{2}})+36{{b}^{4}}\]
This can be re-written as,
$\begin{align}
  & {{\left( 5{{a}^{2}}+6{{b}^{2}} \right)}^{2}}=25{{a}^{4}}+2\left( 5{{a}^{2}} \right)\left( 6{{b}^{2}} \right)+36{{b}^{4}} \\
 & \Rightarrow {{\left( 5{{a}^{2}}+6{{b}^{2}} \right)}^{2}}=25{{a}^{4}}+60{{a}^{2}}{{b}^{2}}+36{{b}^{4}} \\
\end{align}$
Hence the square of polynomial $\left( 5{{a}^{2}}+6{{b}^{2}} \right)$ is $25{{a}^{4}}+60{{a}^{2}}{{b}^{2}}+36{{b}^{4}}$.

Note: Students generally confuse themselves while using formula ${{\left( x+y \right)}^{2}}$ which is equal ${{x}^{2}}+{{y}^{2}}+2xy.$They should also be careful about putting the values and doing the needful calculations.
Another approach for finding the square of the polynomial is to use the direct formula, i.e.,
${{\left( x+y \right)}^{2}}={{x}^{2}}+{{y}^{2}}+2xy$
Here substitute x as $5{{a}^{2}}$ and y as $6{{b}^{2}}$ to get the result.
Using this approach you will get the same answer in much less time.