Addition ,Subtraction ,Multiplication And Division of algebraic expressions

Addition of algebraic expressions (or) polynomials
    Addition of polynomials means adding the like terms of the polynomials 
Note :  Unlike terms can’t be added
1.    The sum of like terms is defined as a like terms similar to each one of them whose coefficent of given terms
    Ex : 7x + 3x - 2x = (7+3-2) x = 8x
2.    Combining the co-efficents like terms of an expression through addition or subtraction is called simplication of an algebraic expression 
    There are ‘2’ methods of adding algebraic expressions they are
    1) Horizontal method    2) Vertical method
Horizontal method : 
     In this method like terms should be added and unlike terms should be written separately by using associative law of addition 
Ex :     Add 8x +7y and 3x - 2y
SOl :         8x + 7y + 3x - 2y 
            = (8x + 3x) + (7y - 2y)
            = 11x + 5y
Vertical method :
     Step I : In this method exprssions to be added and written one below the other.
    Step II : The like terms of each type are placed in separate columns 
    Step III :  The sum will be written below that column 
    Step IV : If a particular like terms is absent in an expression, the place is left vacant 
Ex :
 

Subtraction of Algebraic expressions the additive inverse of number
* The additive inverse of any number is obtainde by a simple change  of is sign, so additive inverse of a number is also called the negative of the number 
Ex : 1) Additive invers of 3/2 is-3/2
             2) Additive inverse of  is -i.e.\(-(-\sqrt3)\) i.e.,\(\sqrt{3}\)
 Additive inverse of an expression :
* The additive inverse (or) the negative of an expression is obtained by replacing each term of the expression with its additive inverse
Ex : 1) Additive inverse of -90x is - (-90x) i.e 90x
* To subtract 1st expression from the second expression. Additive inverse of the 1st expression shoud be addded to the second expression if P and Q are two algebraic expressions  then P - Q = P+(-Q)
 Ex : Subtract 11a - 6b from 7a +4b
Sol :  (7a +4b) - (11a -6b)
    = 7a -4b -11a+6b
    = (7a +11a) +(4b +6b)
    = -4a +10b 
Subtraction also can be done in ‘2’ ways 
* Horiontal method : 
 Ex : Horizontal Method:
Ex : (2x + 3y) - (y - 3x)
     = 2x+3y-y+3x
     = (2x +3x) +(3y-y)
     = 5x + 2y
Vertical Method :  

Multiplication of polynomials :
    Multiply each term of the first polynomial with each term of the second and add the like terms in the poroduct 
    Suppose (a+b) and (c+d) are two polynomials by using the distributive law of multiplication over additiion we can find the product as given below :
    (a+b) x (c+d) = a x (c+d) +b x(c+d)
                = a x c + a x d + b x c + b x d
                = ac + ad + bc +bd
Note : The product of the ‘2’ factors with the same sign is postive and the product of the factors with the opposite sign is negative.
Column and method of multiplication :
     In this method we write multiplicand and the multiplier in descending powers of the valiable arragne one under another, and multiply the multiplicand by every term of the multiplier and add 
Ex :     2x² - 3x + 4 
            3x² - 2x + 1
            ___________
            6x⁴ -9x³  + 12x²
                      - 4x³ + 6x² - 8x
                     + 2x²  - 3x      +4
            ______________________
            6x⁴ - 13x³ + 20x² - 11x + 4
Division of multinomials :  
1)    Arrange the terms of the dividend and divisor in decreasing order of powers keeping zero for missing terms.
2)    Divide the first term of the dividend by the first term of the divisor and write the result as the first term of the quotient
3)    Multiply the entire divisor by this first term of the Quotient and put the product under the dividend, keeping like terms under  each other
4)     Subtract the product from the dividend and bring down the rest of the dividend
5)    Step 4 gives us the now dividend repeat setps 1 to 4
6)     Continue the process till the degree of the remainder becomes zero or less than that of the divisor 
Ex : a) \(4{{x}^{4}}\div x\,\,\,=\,\,4{{x}^{4}}\times \frac{1}{x}\,\,\,=\,\,\,4\left( \frac{{{x}^{4}}}{{{x}^{1}}} \right)=\,\,\,4{{x}^{4-1}}=\,\,4{{x}^{3}}\)
    
Dividing  monomial by a monomial
EX : Divide 35x²y³  by -5x³y<sup>2

In actural working you may combine steps 2 and 3 and arragne your work as under  \(35{{x}^{2}}{{y}^{3}}\div \left( -5{{x}^{3}}{{y}^{2}} \right)=\frac{35{{x}^{2}}{{y}^{3}}}{-5{{x}^{3}}{{y}^{2}}}\)</sup>=\(-7y\over x\)