ALGEBRA BASICS

RULES & PROCEDURE OF ADDITION, SUBTRACTION, MULTIPLICATION & DIVISION IN ALGEBRA

1. Any number, variale, term or expression, which has no sign before it, is positive and for various algebric operations a + sign is always presumed before it.

2. Multiplication or Division by 1: If number, variable, term or expression is multiplied by 1 or divided by 1, the number, variable, term or expression, as the case may be, remains the same. Number, variable, term or expression is not affected on being multiplied or divided by 1.

3. For adding two quantitites consisting numbers, variables or terms, we put a + sign in between the two.

4. For subtractiing a number, variable, term or expression from any quantity, first we put quantity (number, variable, term or expression) from which number, variable, term or expression is to be subtracted and then put - sign and thereafter, we put within a small bracket the number, variable, term or the expression which is to be subtracted. Thereafter we simplify the new expression in the manner given in the last of this page.

5. ADDITION OF AN EXPRESSIONS TO ANY QUANTITY: When an expression is to be added to any quantity, first we put the quantity to which expression is to be added, then a + sign after the expression in which another expression is to be added and thereafter, we put within small brackets the expression which is to be added. Thereafter, we simplify the new expression by adding similar terms and if similar terms of opposite signs then by subtracting smaller term from greater term.
If there are more than two expressions, we follow the same procedure for every expression. We mean to say that leaving first expression, we put each expression within a small bracket and put therese expressions in a row one by one after inserting a + sign before each of the small bracket. Though, in case of addition, it is not necessary to put the expression within small brackets yet I would like to advise to put each expression within a small bracket as precautionery measure.

6. SUBTRACTION OF AN EXPRESSION FROM OTHER QUANTITY: When an expression is to be subtracted from any quantity, first we put the quantity from which expression is to be subtracted, then we put a - sign before the expression to be subtracted and thereafter the expression within a small bracket. In case of subtraction it is absolutely necessary to put the expression to be subtracted within a small bracket.

In algebra, when we multiply two terms, we get a single term. While multiplying a term by another term we follow following principles:

coefficient of first term and coefficient of second term are multiplied to give coefficient of resultant term.

any variable of first term gets multiplied with same variable of second term irrespective of their powers. In the resultant term, power of any variable is the sum of powers of such variable in first term and second term. If one term has a variable which other term does not have then such variable is put as such in the resultant term.

For multiplication of variables, it is not necessary that powers of a variable in two terms should be same. For example: If we multiply xy² by the term x²y³ we get x¹⁺²y²⁺³ that is x³y⁵. Here we see that x and y are two variables. In first term x has power of 1 whereas in second term x has power of 2. Similarly in case of variable y, it has power of 2 in first term and power of 3 in second term. On multiplication x gets multiplied by x² to give x¹⁺² or x³. Similarly, y has power of 2 in first term and power of 3 in second term, on their multiplication they give y²⁺³ or y^5.

7. NO SIGN BEFORE SMALL BRACKET CONTAINING AN EXPRESSION: When a sign is missing before any bracket having any expression, it is understood that there exists a + sign before the bracket.

8. SIGN BEFORE SMALL BRACKET CONTAINING AN EXPRESSION: When a sign occurs before any bracket having any expression, while removing small bracket, sign of every term of the expression is multiplied by the sign which existed before the bracket.

10. USE OF MULTIPLICATION SYMBOL IN ALGEBRA: In Algebra, for multiplication we use sign *. For example 2x * 3 is 2x*3 or 6x. In case of variables of the same term, multiplication sign is not written. It is customery. In case of two variables written together without any sign in between them, a * sign is always understood to be there. For example: xy is equivalent to x * y. In between a coefficient and rest part of the term, a sign of * is presumed because coefficient tells how many times a variale is.

11. MUTIPLICATION OF TWO EXPRESSIONS: We know that an expressions generally has more than one term. In between any two terms of an expression, there exists a + or - sign. While multiplying one expression to the other, each term of one expression is multiplied by each term of the other expression and terms obtained are put in a row with their + or - signs. Finally similar terms are added or subtracted (if both similar terms are of the same sign, then they to be added and if similar terms of opposite sign, then smaller is to subtracted from the bigger) to simlify the resultant expression.

First we multiply each term of first expression 2x +3y +z by first term x of second expression and then we multiply each term of first expression by second term 2y of the second expression. What we get is

We have multiplied each term of first expression by each term of second expression by taking terms of second expression one by one. We could have achieved the same result, if we have multiplied each term of second expression by each term of first expression by taking terms of first expression one by one. Usually we multiply all terms of larger expression ( expression having large number of terms) by each term of smaller expression (expression having lesser number of terms).

Division by a - sign affects sign of each term in the denominator. Sign of every term gets reversed. If sign of any term in numerate is +, after division by - sign, the resultant sign of the term is - and if sign of term is -, then after division by - sign, the sign of resultant term is +. How does it happen? In multiplication of + & - signs we have seen that + * - = - & - * + = - and - * - = +. So when we divide - sign by - sign we divide + * - by - or - * + by -. In this case - sign from the numerator is cancelled by the - sign in the denominator and + sign is left. When we divide + sign in numerator by a - sign in denominator, we divide - * - by -. Here one - in numerator gets cancelled by one - sign in denominator and resultant - sign remains.

Since + * + = +, therefore, division of + sign by a + sign does not affect sign of the term to be divided. If denominator has a - sign, then we know that a + sign can be thought of - * - sign. But this can again be written as - * (+ * -) because breaking second - sign we can write +*- in place of it. If we divide - * + * - by a - sign, we are left with -*+ which gives us a - sign.

14. DIVISION SIGN ÷ : Normally, we use a slash / sign in place of a ÷ sign. While performing any operation on any term or expression, we convert sign of ÷ or / into a * sign. For doing this we interchange numerator and denominator of the term which appers after ÷ sign. If term does not have its denominator then denominator is taken as 1. For example: 3x ÷ 5y is 3x * 1/5y which is 3x/5y. Any term whose denominaotr has not been shown, it always has 1 as its denominator. That is why inverse of 5y has been written as 1/5y.

4. In next step, we remove the small bracket. For removing the small bracket, we multiply sign of each term of expression within the bracket by - sign and put each term with its resultant sign. One can also say that + sign within the bracket is changed to - sign and - sign within the bracket is changed to a + sign.

Now we will remove small bracket by multiplying each term within it by - sign because before bracket there is a - sign. This will give us:

3x² + 3x + 5y + 7z - 9x -5y - 3z ( on multiplying signs within bracket by - sign, sign of each term within the bracket has been changed to - sign. (because - * + = -).

Now we see that 3x & 9x are similar terms having x variable and 5y & 5y are similar terms. we simplify them (we add similar terms if both of them are of same sign and we subtract smaller term from larger similar term if both are of opposite signs).

RULES & PROCEDURE OF ADDITION, SUBTRACTION, MULTIPLICATION & DIVISION IN ALGEBRA
1. Any number, variale, term or expression, which has no sign before it, is positive and for various algebric operations a + sign is always presumed before it.
2. Multiplication or Division by 1: If number, variable, term or expression is multiplied by 1 or divided by 1, the number, variable, term or expression, as the case may be, remains the same. Number, variable, term or expression is not affected on being multiplied or divided by 1.
3. For adding two quantitites consisting numbers, variables or terms, we put a + sign in between the two.
4. For subtractiing a number, variable, term or expression from any quantity, first we put quantity (number, variable, term or expression) from which number, variable, term or expression is to be subtracted and then put - sign and thereafter, we put within a small bracket the number, variable, term or the expression which is to be subtracted. Thereafter we simplify the new expression in the manner given in the last of this page.
5. ADDITION OF AN EXPRESSIONS TO ANY QUANTITY: When an expression is to be added to any quantity, first we put the quantity to which expression is to be added, then a + sign after the expression in which another expression is to be added and thereafter, we put within small brackets the expression which is to be added. Thereafter, we simplify the new expression by adding similar terms and if similar terms of opposite signs then by subtracting smaller term from greater term. If there are more than two expressions, we follow the same procedure for every expression. We mean to say that leaving first expression, we put each expression within a small bracket and put therese expressions in a row one by one after inserting a + sign before each of the small bracket. Though, in case of addition, it is not necessary to put the expression within small brackets yet I would like to advise to put each expression within a small bracket as precautionery measure.
6. SUBTRACTION OF AN EXPRESSION FROM OTHER QUANTITY: When an expression is to be subtracted from any quantity, first we put the quantity from which expression is to be subtracted, then we put a - sign before the expression to be subtracted and thereafter the expression within a small bracket. In case of subtraction it is absolutely necessary to put the expression to be subtracted within a small bracket.
MULTIPLICATION
In algebra, when we multiply two terms, we get a single term. While multiplying a term by another term we follow following principles:
sign	sign of first term and sign of second term are multiplied to give sign of resultant term.
coefficient	coefficient of first term and coefficient of second term are multiplied to give coefficient of resultant term.
variable	any variable of first term gets multiplied with same variable of second term irrespective of their powers. In the resultant term, power of any variable is the sum of powers of such variable in first term and second term. If one term has a variable which other term does not have then such variable is put as such in the resultant term.
Powers of variable	For multiplication of variables, it is not necessary that powers of a variable in two terms should be same. For example: If we multiply xy² by the term x²y³ we get x¹⁺²y²⁺³ that is x³y⁵. Here we see that x and y are two variables. In first term x has power of 1 whereas in second term x has power of 2. Similarly in case of variable y, it has power of 2 in first term and power of 3 in second term. On multiplication x gets multiplied by x² to give x¹⁺² or x³. Similarly, y has power of 2 in first term and power of 3 in second term, on their multiplication they give y²⁺³ or y^5.
7. NO SIGN BEFORE SMALL BRACKET CONTAINING AN EXPRESSION: When a sign is missing before any bracket having any expression, it is understood that there exists a + sign before the bracket.
8. SIGN BEFORE SMALL BRACKET CONTAINING AN EXPRESSION: When a sign occurs before any bracket having any expression, while removing small bracket, sign of every term of the expression is multiplied by the sign which existed before the bracket.
9. MULTIPLICATION OF SIGNS:
Signs to be multiplied	Product of signs
+ * +	+
*- -**	+
*+ -**	-
*- +**	-
We see that on multiplication of same signs ( either both + or both -) we get a + sign and on multiplication of different signs (+ * - or - * + that is one + and other -), we get - sign.
10. USE OF MULTIPLICATION SYMBOL IN ALGEBRA: In Algebra, for multiplication we use sign . For example 2x 3 is 2x3 or 6x. In case of variables of the same term, multiplication sign is not written. It is customery. In case of two variables written together without any sign in between them, a sign is always understood to be there. For example: xy is equivalent to x * y. In between a coefficient and rest part of the term, a sign of * is presumed because coefficient tells how many times a variale is.
11. MUTIPLICATION OF TWO EXPRESSIONS: We know that an expressions generally has more than one term. In between any two terms of an expression, there exists a + or - sign. While multiplying one expression to the other, each term of one expression is multiplied by each term of the other expression and terms obtained are put in a row with their + or - signs. Finally similar terms are added or subtracted (if both similar terms are of the same sign, then they to be added and if similar terms of opposite sign, then smaller is to subtracted from the bigger) to simlify the resultant expression.
EXAMPLE: Find the product of 2x +3y +z and x +2y.
First we multiply each term of first expression 2x +3y +z by first term x of second expression and then we multiply each term of first expression by second term 2y of the second expression. What we get is
x2x + x3y +xz + 2y2x +2y3y + 2yz
Or 2x² + 3xy + xz + 4xy + 6y² + 2yz
There are two similar terms 3xy and 4xy and both have + sign, therefore on adding them, we get-
2x² + 7xy + xz + 6y² + 2yz
Or 2x² + 6y² + 7xy + xz + 2yz
We have multiplied each term of first expression by each term of second expression by taking terms of second expression one by one. We could have achieved the same result, if we have multiplied each term of second expression by each term of first expression by taking terms of first expression one by one. Usually we multiply all terms of larger expression ( expression having large number of terms) by each term of smaller expression (expression having lesser number of terms).
12. DIVISION OF + or - SIGN IN NUMERATOR BY - SIGN IN DENOMINATOR:
Division by a - sign affects sign of each term in the denominator. Sign of every term gets reversed. If sign of any term in numerate is +, after division by - sign, the resultant sign of the term is - and if sign of term is -, then after division by - sign, the sign of resultant term is +. How does it happen? In multiplication of + & - signs we have seen that + * - = - & - * + = - and - * - = +. So when we divide - sign by - sign we divide + * - by - or - * + by -. In this case - sign from the numerator is cancelled by the - sign in the denominator and + sign is left. When we divide + sign in numerator by a - sign in denominator, we divide - * - by -. Here one - in numerator gets cancelled by one - sign in denominator and resultant - sign remains.
Examples:
Term	is the same as
-x / -y	+ * - x / -y or x/y
x/ -y	- *- x /y or -x/y
2x /-3	-*- 2x/-3 Or -2x/3
(2x +3y) /-1	-2x -3y
13. DIVISION OF + or - SIGN IN NUMERATOR BY + SIGN IN DENOMINATOR:
Since + * + = +, therefore, division of + sign by a + sign does not affect sign of the term to be divided. If denominator has a - sign, then we know that a + sign can be thought of - * - sign. But this can again be written as - * (+ * -) because breaking second - sign we can write +- in place of it. If we divide - + * - by a - sign, we are left with -*+ which gives us a - sign.
14. DIVISION SIGN ÷ : Normally, we use a slash / sign in place of a ÷ sign. While performing any operation on any term or expression, we convert sign of ÷ or / into a * sign. For doing this we interchange numerator and denominator of the term which appers after ÷ sign. If term does not have its denominator then denominator is taken as 1. For example: 3x ÷ 5y is 3x * 1/5y which is 3x/5y. Any term whose denominaotr has not been shown, it always has 1 as its denominator. That is why inverse of 5y has been written as 1/5y.
15. SUBTRACTION OF ONE EXPRESSION FROM ANOTHER EXPRESSION:
For subtracting one expression from another expression, following steps are taken:
1. First we put expression from which another expression is to be subtracted.
2. Then we put a - sign
3. Thereafter, we put the expression, which is to be subtracted, within a small bracket.
4. In next step, we remove the small bracket. For removing the small bracket, we multiply sign of each term of expression within the bracket by - sign and put each term with its resultant sign. One can also say that + sign within the bracket is changed to - sign and - sign within the bracket is changed to a + sign.
5. In the next step, we simplify the expression by adding or subtracting (as may be required) similar terms.
Example: Subtract 9x +5y + 3z from 3x² + 3x + 5y + 7z
Here expression to be subtracted is 9x +5y + 3z and expression from which it is to be subtracted is 3x² + 3x + 5y + 7z. So we will write:
3x² + 3x + 5y + 7z - (9x +5y + 3z)
Now we will remove small bracket by multiplying each term within it by - sign because before bracket there is a - sign. This will give us:
3x² + 3x + 5y + 7z - 9x -5y - 3z ( on multiplying signs within bracket by - sign, sign of each term within the bracket has been changed to - sign. (because - * + = -).
Now we see that 3x & 9x are similar terms having x variable and 5y & 5y are similar terms. we simplify them (we add similar terms if both of them are of same sign and we subtract smaller term from larger similar term if both are of opposite signs).
On simplification, we get 3x² -6x -3z
16. ONLY SIMILAR TERMS (OR EXACTLY EQUIVALENT OR SAME TERMS) MAY BE ADDED OR SUBTRACTED TO GIVE A SINGLE TERM. OTHER TYPE OF TERMS CANNOT BE ADDED OR SUBTRACTED TO GIVE A SINGLE TERM.
17. TWO TERMS CAN ALWAYS BE MULTIPLIED OR DIVIDED TO GIVE A SINGLE TERM.
TO BE CONTINUED ...