Question

Add the following polynomials
$\left( i \right) - 3{a^2}{b^2},\dfrac{{ - 5}}{2}{a^2}{b^2},4{a^2}{b^2},\dfrac{2}{3}{a^2}{b^2}$
$\left( {ii} \right)\left( {3{x^2} + 7xy - 6{y^2}} \right),\left( {4{x^2} - 3xy + 2{y^2}} \right),\left( { - a{x^2} + xy - {y^2}} \right)$

Answer 1

Hint – In this particular question use the concept that if there are like terms (i.e. the polynomial having same degree and have same variables for example ${x^2}$ and $2{x^2}$) are simple added and give a term which have same degree but the coefficient is either negative or positive depending upon the coefficients of the polynomials so use these concepts to reach the solution of the question.

Complete step-by-step answer:
We have to add the given polynomials
$\left( i \right) - 3{a^2}{b^2},\dfrac{{ - 5}}{2}{a^2}{b^2},4{a^2}{b^2},\dfrac{2}{3}{a^2}{b^2}$
As we see that in the above all given polynomials have the same degree in a as well as in b so these all are like terms so it is all simply added and gives a resultant polynomial which also have the same degree in a as well as in b.
If these are unlike terms (i.e. degree of a is different in one polynomial and degree of a is different in another polynomial same case for b) so when we added these terms they will not give us a resultant term it will give us simply the addition value for example the sum of 2x and ${x^2}$is $\left( {2x + {x^2}} \right)$.
So the addition of the above polynomial is
 $ \Rightarrow - 3{a^2}{b^2} + \dfrac{{ - 5}}{2}{a^2}{b^2} + 4{a^2}{b^2} + \dfrac{2}{3}{a^2}{b^2}$
Now take ${a^2}{b^2}$ common from all the terms we have,
\[ \Rightarrow {a^2}{b^2}\left( { - 3 + \dfrac{{ - 5}}{2} + 4 + \dfrac{2}{3}} \right)\]
Now simplify these additions by taking the LCM we have,
\[ \Rightarrow {a^2}{b^2}\left( {\dfrac{{ - 3\left( 6 \right) - 5\left( 3 \right) + 4\left( 6 \right) + 2\left( 2 \right)}}{{2\left( 3 \right)}}} \right)\]
\[ \Rightarrow {a^2}{b^2}\left( {\dfrac{{ - 18 - 15 + 24 + 4}}{6}} \right)\]
\[ \Rightarrow \dfrac{{ - 5}}{6}{a^2}{b^2}\]
So this is the required sum.
$\left( {ii} \right)\left( {3{x^2} + 7xy - 6{y^2}} \right),\left( {4{x^2} - 3xy + 2{y^2}} \right),\left( { - a{x^2} + xy - {y^2}} \right)$
In this equation like degree polynomials having same variables are added together so we have,
$ \Rightarrow \left( {3{x^2} + 7xy - 6{y^2}} \right) + \left( {4{x^2} - 3xy + 2{y^2}} \right) + \left( { - a{x^2} + xy - {y^2}} \right)$
$ \Rightarrow \left( {3 + 4 - a} \right){x^2} + \left( {7 - 3 + 1} \right)xy + \left( { - 6 + 2 - 1} \right){y^2}$
$ \Rightarrow \left( {7 - a} \right){x^2} + 5xy - 5{y^2}$
So this is the required sum.

Note – Whenever we face such types of questions the key concept we have to remember is that always remember the difference between the like and unlike terms which is all stated above so first check in the given polynomials which are like terms and which are unlike terms so after this simply add them we will get the required answer.