Question

Add the following polynomials, ${{x}^{2}}-{{y}^{2}}-1,{{y}^{2}}-1-{{x}^{2}},1-{{x}^{2}}-{{y}^{2}}$.

Answer 1

Hint: To solve this question, we should know that we have to add the polynomials of the same degree with the same variables and not all the different terms together. For example, when we add, ${{x}^{2}}+1+{{x}^{2}}+2$, we will get $2{{x}^{2}}+3$. We will follow this concept to get the required answer.

Complete step-by-step answer:

It is given in the question that we have to add the given polynomials, ${{x}^{2}}-{{y}^{2}}-1,{{y}^{2}}-1-{{x}^{2}},1-{{x}^{2}}-{{y}^{2}}$. We know that we can add only the polynomials of the same degree and with the same variable, and similarly constants are added with the constants. So, a term with ${{x}^{2}}$ is added with another term that includes ${{x}^{2}}$ and ${{y}^{2}}$ will be added with ${{y}^{2}}$ terms and the constants will be added with the constants only. We have been given three polynomials. So, we will add the first two polynomials and add the answer of that with the
third polynomial. So, we will add ${{x}^{2}}-{{y}^{2}}-1,{{y}^{2}}-1-{{x}^{2}}$. So, we get,
$\begin{align}
  & ={{x}^{2}}-{{y}^{2}}-1+{{y}^{2}}-1-{{x}^{2}} \\
 & \Rightarrow {{x}^{2}}-{{x}^{2}}-{{y}^{2}}+{{y}^{2}}-1-1 \\
 & \Rightarrow -2 \\
\end{align}$
Now, we will add - 2 with the third polynomial. So, we will get as,
$\begin{align}
  & =\left( -2 \right)+\left( 1-{{x}^{2}}-{{y}^{2}} \right) \\
 & \Rightarrow -2+1-{{x}^{2}}-{{y}^{2}} \\
 & \Rightarrow \left( -1-{{x}^{2}}-{{y}^{2}} \right) \\
\end{align}$
Now, we will multiply the obtained polynomial with -1. So, we will get, -$\left( {{x}^{2}}+{{y}^{2}}+1 \right)$. Therefore, we get the sum of the three given polynomials as, -$\left( {{x}^{2}}+{{y}^{2}}+1 \right)$.

Note: Most of the students make a mistake with the signs while doing the addition of the polynomials. Sometimes the students get confused with the terms and may add $\left( {{x}^{2}}+{{y}^{2}} \right)$ as ${{\left( x+y \right)}^{2}}$, which is incorrect as that is equal to ${{x}^{2}}+{{y}^{2}}+2xy$. Therefore, it is advisable that this type of questions should be solved carefully by paying attention to the signs and the basic concepts of addition of polynomials.