1. ADDITION :- The sum of coefficients of like terms is performed.
For eg. :- Add $41x^{2}\,-6xy\,+12y^{2}\;\;and\;\;11x^{2}\,+2xy\,\,-7y^{2}$$\begin{align*}41x^{2}\,-6xy\,+12y^{2}\\+\, \: 11x^{2}\,+2xy\,\,-7y^{2}\ \\ \hline52x^{2}\, -4xy\,\, +5y^{2}\ \\ \hline\end{align*}$
2. SUBTRACTION :- The difference of coefficients of like terms is performed.
Rule :- Change the sign of each term of the expression to be subtracted and add it to the expression from which subtraction was to be made.
For eg. :- Subtract $5x^{2}\, -9xy\, -3y^{2}\;\: from\; \: 3x^{2}\, -5xy\, +2y^{2}$
$\begin{align*}3x^{2}\, -5xy\, +2y^{2}\\(-)\; \ \: 5x^{2}\, -9xy\, -3y^{2}\\-\; \; \; \; \; \; +\; \; \; \; \; \; \; +\; \; \; \; \! \; \;\\\hline-2x^{2}\: +4xy\: +5y^{2}\\\hline\end{align*}$
3. MULTIPLICATION :- Multiply constants as we perform multiplication of rationals and multiply like variables by using the laws of exponents.
For eg. :- Multiply $-5x^{3}y,\; 7xy\; \; and\;\; 3x^{2}y^{2}$
$(-5x^{3}y)\;\times (7xy)\;\times (3x^{2}y^{2})$
$\Rightarrow\;\;\left[(-5)\times(7)\times(3)\right](x^{3}\times x\times x^{2})(y\times y\times y^{2})$
$\Rightarrow\;\;-105x^{6}y^{4}$
4. DIVISION :-
For eg. :-
(1.) Divide $108x^{3}y^{3}z^{7}\; \; by\; \; 120x^{2}y^{2}z^{2}$
Sol. $\begin{align*}108\; x^{3}y^{3}z^{7}\\\hline120\; x^{2}y^{2}z^{2}\end{align*}$
$\Rightarrow\; \; \frac{108}{120}\; \frac{x^{3}}{x^{2}}\; \frac{y^{3}}{y^{2}}\; \frac{z^{7}}{z^{2}}$
$\Rightarrow \; \; \frac{9}{10}xyz^{5}$
(2.) Divide $4x^{2}-8x+3\; \; by\; \; 2x-1$
Sol.
Quotient = 2x - 3
Remainder = 0
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