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Multiplication of algebraic expressions
    Following rules of signs and the laws of exponents used in multiplication.

1.     Rules of signs in multiplication : 
    The product of two factors with like signs is positive, and the product of two factors with unlike signs is negative. Thus if a and b are two positive numbers then 

2.    Laws of exponents in multiplication 
       Law of exponents in multiplication is given as :
        am x an  = am+n  x e.g. x5 x x2 = x5 + 2 = x7 

Multiplication of a Monomial by a Monomial : 

      Thus in multiplication of algebraic expression : 
    (i)     Write the product of the numerical coefficients. 
    (ii)     Write all the different letters occurring in the algebraic expressions giving to each letter an exponent (power) equal to the sum of all the exponents of that letter in the given expressions. 
    (iii)     The sign of the product is minus if there is an odd number of negative factors and plus if there is an even number of negative factors.

Multiplication of a Binomial by a Monomial : 
    In order to multiply a binomial by a monomial use the following rule : 
    a x (b + c)  = a x b + a x c. 

    Illustration 
        Multiply  : (i)   3x2 + 4xy by 2x         (ii) –4a [a + 3b]    (iii) – 4a [a – 3b]
    Solution    
        (i)     2x [3x2 + 4xy] = 2x x 3x2 + 2x x 4xy  = 6x3 + 8x2
        (ii)     – 4a [a + 3b]  = – 4a x a  – 4a x 3b = – 4a2 – 12ab 
        (iii)     – 4a [a – 3b] = – 4a ´ a – 4a ´ (–3b) = – 4a2 + 12 ab. 

Multiplication of a Binomial by a Binomial : 
    In multiplication of a binomial by binomial we will use the law of multiplication of a binomial by a monomial twice. 
    Illustration     
        (i)     (a + b) (c + d) 
        (ii)     (2x2 + 3y) (3x2 – 2y)
    Solution
         (i)     (a + b) (c + d)     = a (c + d) + b (c + d)
                      = a x c + a x d + b x c + b x d
                     = ac + ad + bc + bd 

        (ii)     (2x2 + 3y) (3x2 – 2y) 
                    = 2x2 (3x2 – 2y) + 3y (3x2 – 2y)
                    = 2x2 x 3x2 + 2x2 x (– 2y) + 3y x 3x2 + 3y x (–2y) 
                    = 6x4 – 4x2y + 9x2y – 6y2

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