MULTIPLICATION & DIVISION OF ALGEBRAIC EXPRESSION OR POLYNOMIALS

MULTIPLICATION & DIVISION OF ALGEBRAIC EXPRESSION OR POLYNOMIALS
MULTIPLICATION OF ALGEBRAIC EXPRESSIONS
Multiplication of algebraic expressions involves the following steps.
    Multiply the coefficients of the terms, add the powers of the variables with the same base, and obtain the algebraic sum of the like and unlike terms. Before learning about the multiplication of algebraic expressions, let’s see some of the algebraic rules for multiplication.
    i) The product of two terms with the same signs is positive, and the product of two terms with different signs is negative.
For example :
    . -a x -b = ab
    . a x - b = -ab
    ii) We can add the exponents of the same bases in case of multiplication as a^m x aⁿ = a^m+n
Example : Find the product of 6ab and -3a²b³
Solution : 6ab x -3a²b³
        = 6 x ^-3 x ab x a²b³
        = -18 x a¹⁺² x b¹⁺³
        = -18a³b⁴
1)    Multiplication of a Polynomial by a Monomial
An algebraic expression is considered a polynomial when it contains variables, coefficients, that involve only the operations of subtraction, addition, multiplication, and non-negative integer exponentiation of variables.
    Multiply each term of the polynomial by the monomial, using the distributive law : a x (b + c) = a x b + a x c
Example : Find following product : \(5{a^2}{b^2} \times \left( {3{a^2} - 4ab + 6{b^2}} \right)\)
Solution:    
    \(\begin{gathered} = \left( {5{a^2}{b^2}} \right) \times \left( {3{a^2}} \right) + \left( {5{a^2}{b^2}} \right) \times \left( { - 4ab} \right) + \left( {5{a^2}{b^2}} \right) \times \left( {6{b^2}} \right) \hfill \\ = 15{a^4}{b^2} - 20{a^3}{b^3} + 30{a^2}{b^4} \hfill \\ \end{gathered} \)
2)    Multiplication of Two Binomials 
    An algebraic expression is considered binomial when it is made of the sum or difference of two terms. We multiply two binomials by using the distributive law of multiplication twice.

Example : Multiply (3a + 5b) and (5a - 7b)
Solution : (I) Horizontal multiplication method 
            (3a + 5b) x (5a - 7b)
            = 3a x (5a - 7b) + 5b x (5a - 7b)
            = (3a x 5a - 3a x 7b) + (5b x 5a - 5b x 7b)
            = (15a² - 21ab) + (25ab - 35b²)
            = 15a² - 21ab + 25ab - 35b²
            = 15a² +4ab-35b²
(II) Column wise multiplication  
        3x + 5y
            x (5x - 7y)
        15x² + 25xy       \(\to\)    Multiplication by 5x.
         -21xy - 35y²       \(\to\)     Multiplication by -7y.
        15x² + 4xy - 35y²   \(\to\)  Added the above terms.
3)    Multiplication by Polynomial 
Example : Multiply  \(\left( {5{x^2} - 6x + 9} \right)with\left( {2x - 3} \right)\)
Solution : 
        x (2x - 3 )
          10x³ - 12x² + 18x   \(\to\)  multiplication by 2x
           -15x² + 18x - 27    \(\to\)   multiplication by -3
              10x³ - 27x² + 36x -27 \(\to\) added the above terms
Therefore  \(\left( {5{x^2} - 6x + 9} \right) \times \left( {2x - 3} \right)\) is \({\text{ }}\left( {{\text{10}}{{\text{x}}^{\text{3}}}{\text{ - 27}}{{\text{x}}^{\text{2}}}{\text{ + 36x - 27}}} \right)\)