Guide

In multiplying a monomial and a polynomial, we use the distributive property as well as the rules in multiplying monomials which we have just reviewed in the preceding lesson. 

  Study the following examples

In multiplying polynomials, the distributive property and the laws of exponents are used extensively. Observe how these two concepts are used in the following illustrations.

       1. 2x(x + y – 2) = 2x(x) + 2x(y) + 2x(-2)             Distributive property

                         = 2x2 + 2xy – 4x                 am an = am+n that is, 

2.   3a2(a2 + 2ab + 3b2 – 3) = 3a2(a2) + 3a2(2ab) + 3a2(3b2) + 3a2(-3)

                                = 3a4 + 6a3b + 9a2b2 – 9a2

      3. (m + 5)(m – 2)  =  m(m – 2) + 5(m – 2)            Distributive Property

                     = m2 – 2m + 5m – 10            Again Distributive Property

                                        = m2 + 3m – 10               Adding Similar Terms

4.   (3a – 3)(2a + 4)      = 2a(3a – 3) + 4(3a – 3)

                                         = 6a2 – 6a + 12a – 12            

                                         = 6a2 + 6a - 12       

FOIL Method

in finding the product of terms in the binomials used these words: First Terms, Outside Terms, Inside Terms, and Last Terms. This is called the FOIL Method for finding the product of two binomials.

Illustrative Example 1

                                  First        Outer         Inner        Last

( x + 4 )  ( x + 2) = ( x · x )  +   (2 · x)   +   (4 · x)   +   (4 · 2)

                              = x2 + 2x + 4x + 8

                              = x2 + 6x + 8 

Illustrative Example 2

(2m5 + m2) ( -m3 + m) =   2m5   ·  (-m3) + 2m5   ·  (m)  +  m2  ·  (-m3)  + m2     ·   m

                                       = -2m8  + 2m6 -  m5  + m3

PRODUCT OF ( a + b ) and ( a – b )

                 The product of the sum and the difference of two terms is equal to the       square of the first term minus the square of the second term. That is,

                 ( a + b) ( a – b ) = a2 – b2

            The product is called the difference between two squares.

Illustrative Example 1

( 3x + 4y) ( 3x – 4y ) =  9x2 – 16 y2

Illustrative Example 2

( 5 + 8a2 )( 5 – 8 a2 ) =  25 – 64 a4 

SQUARE OF A BINOMIAL

        The square of a binomial is equal to the square of the first term, plus or minus twice the product of the two terms, plus the square of the last term.

That is:

       ( a   +   b )2          =  a2  +        2 ( ab)        +   b2

      First term                   (1st term) 2 + 2 (1st term  · 2nd  term) + ( 2nd term) 2

       Second term

                             = a2 + 2ab + b2

       ( a – b )2           = a2 – 2 ( ab) + b2

                             = a2 – 2 ab + b2

 When you square a binomial the product is a perfect square trinomial.

 Illustrative Example 1

( x + 8 )2  = x2 + 16x + 64

 Illustrative Example 2 

( a – 9 ) 2   = a2 – 18a + 81

DIVISION OF POLYNOMIAL

Use the law of exponents and the definition together with the operations on real numbers in dividing monomials.

       1.  48x3 ÷ 16x2 = 48 x3-2 = 3x    or       3(16)x · x · x  = 3x

                                                       16                             16x  · x                           

       2. -24xy3  ÷ 4x2y5  = -24   ·   1   ·    1    =   -6  

                                     4       x2-1        y5-3             x y2

       3. -27xz2  ÷ 9xz   = 3x1-1 z2-1  =  3x0 z = 3z

To divide a monomial by a monomial:

1. Divide the numerical coefficients. Express the quotient as a rational number in simplest form.

2. Apply the laws of exponent and make all exponents positive.

To divide a polynomial by a monomial

1. Divide each term of the polynomial by the monomial.

2. Add the resulting quotient. Be sure to indicate the sign before each term using the rule of sign.

Note:

1. Before dividing, arrange the terms in descending power of the variables.

2. Insert 0 for the missing terms of the dividend or the divisor.