Addition of Like Terms: Sum of two or more like terms is the like term whose coefficient is equal to the sum of all the numerical coefficient of the given like terms.

$\displaystyle \text{Example: } 2xy+4xy-xy=(2+4-1)xy=5xy$ (you can add like terms)

Subtraction of Like Terms: This also is similar the above operation. The difference of the two like terms is the like terms whose numerical coefficient is the difference of the numerical coefficient of the given like terms.

$\displaystyle \text{Example: }7xy-3xy=(7-3)xy=4xy$

$\displaystyle \text{Or }-7xy-4xy=(-7-4)xy=-11xy$

You could also change the sign of the second term and add this to the first term.

Take the example: If we have to subtract $3xy$ from $7xy$, change $3xy$ to $-3xy$ and add the two terms. The result would be the same.

Two or more polynomials can be added by first arranging them so that the like terms lie in the same vertical column. Then add each column, starting from left to right.

If there are terms that are only in one of the polynomial, you still have to arrange then, though they might be the only term in that column.

$\displaystyle \text{Example: } 4x-7y+z+10; 5x-3y-28; x+11y+3a+15$

$\begin{array}{r r r r r r r r r r } & 4x & - & 7y & + & z & & & + & 10 \\ & 5x & - & 3y & & & & &- & 28 \\ (+) & x & + & 11y & & & + & 3a & + & 15 \\ \hline & 10x & - & y & + & z & + & 3a & - & 3 \end{array}$

Subtraction of Polynomials: Subtraction of polynomials is also very similar to addition, with one difference. The polynomial that is to be subtracted, we need to change the sign and then just add them as shown below in the example.

$\displaystyle \text{Example: Subtract }4x-7y+z+10 \text{ from } 5x-3y-28$

$\begin{array}{r r r r r r r r } & 5x & - & 3y & & & - & 28 \\ (-) & 4x & - & 7y & + & z & + & 10 \\ \hline & x & + & 4y & - & z & - & 38 \end{array}$

Multiplication of a Polynomial

Multiplication of Monomials

Product of Monomials $=$ (product of their numerical coefficients)$\times$ (product of their variable parts)

Let’s do an example.

Multiply $3xy$ and $2x$ which is $(3\times 2)(xy\times x)=6x^2y$

Please note that the laws of exponents would come into picture. So you can refer to the on our website for a refresher.

Multiplication of a Polynomial by a Monomial

Here distributive laws will come into play. Please refer to the Chapter 19 in our blog for another refresher.

Note: Distributive laws:

1. $(a+b)\times c=ac+bc$
2. $(a-b)\times c=ac-bc$
3. $a\times (b+c)=ab+ac$
4. $a\times (b-c)=ab-ac$

Let’s do some examples:

$\displaystyle \text{Example: Multiply } (2x+3y)\times 3x^2y=6x^3y+9x^2y^2$

Multiplication of a Polynomial by a polynomial: In this you multiply each of the multiplicand with each term of the multiplier and take the sum of all the products obtained.

Let’s do another example to simplify the above statement.

$\displaystyle \text{Multiply } (3x^2-5+2x)\times (2x^2+x)$

$\displaystyle \text{First multiply } (3x^2-5+2x)\times (2x^2)$

$\displaystyle \text{and then multiply } (3x^2-5+2x)\times (x)$

and then add the two expressions

$\begin{array}{r r r r r r r r } & 6x^4 & + & 4x^3 & - & 10x^2 & & \\ (+) & & & 3x^3 & + & 2x^2 & - & 5x \\ \hline & 6x^4 & + & 7x^3 & - & 8x^2 & - & 5x \end{array}$

Division of Polynomials

Division of a monomial by another monomial is simple. This is something that we have already done in the exponents.

$\displaystyle \text{Example: Divide } 25x^2y^3 \text{ by } 5xy= \frac{25x^2y^3}{5xy} = 5xy^2$

Division of a polynomial by another monomial is also simple. This is something that we have already done in the exponents.

$\displaystyle \text{Example: Divide } (25x^2y^3+5xy^2) \ by\ 5xy \\ \\ = \frac{25x^2y^3+5xy^2}{5xy} = \frac{25x^2y^3}{5xy} + \frac{5xy^2}{5xy} =5xy^2+y$

Division of a polynomial by another polynomial is a little more complicated.

Step 1: First step is to arrange the dividend and the divisor in decreasing order of their degree.

Step 2: The second step is to multiply the first term of the divisor by such a number that you get the first term of the dividend.

Step 3: Multiply the entire divisor expression by this number and then subtract the obtained expression from the dividend.

Step 4: The remainder that you obtain, consider it the dividend and repeat the process again until we don’t get a remainder.

Example: Divide $x^2-9x-10 \ by\ x+1$

$x+1)\overline{x^2-9x-10}(x-10 \\ \underline{(-)\ \ \ x^2+1} \\ {\hspace{1.6cm} -10x-10} \\ \underline{(-) {\hspace{1.0cm} -10x-10}} \\ {\hspace{2.0cm} 0}$

Step 1: Multiple $x$ by$x$  so that we get $x^2$. This is to multiply $x$ by another number so that we get the first term of the dividend.

Step 2: is to multiple the entire divisor by $x$ . We get $x^2+x$. Then subtract this from the dividend.

Step 3: is to obtain the remainder by subtracting We get $x^2+x$ from $x^2-9x-10$ to get the remainder $-10x-10$

Step 4: We then multiply $x$ by $-10$ to get a term equal to the first term of the remainder. Repeat Step 3 with the new expressions.

The steps will become clearer once we do an example.

Simplification by removal of brackets and use of BODMAS

The standard BODMAS rule applies to algebraic operations. Please refer to chapter for a refresher on BODMAS

If there is a $+$ (plus) sign in from of the bracket, then the brackets are removed without making any change in the sign of the terms inside the brackets. If – (minus) sign is in front of the bracket, then the sign of all the terms inside the brackets need to be reversed.