Polynomials - Part II

2. Binomials 
Addition or subtraction of monomials give rise to binomials. For example, x + 4,  2x + 3y,  5a²+ 10b,  x² ñ y² are binomials.

Binomials can be added, subtracted, multiplied, divided by other binomials or monomials, but each variable has to be treated independently and carefully. Logic would be to try and group like variables together. Sometimes such mathematical operations lead to higher level of equations that are called trinomials, or polynomials in general. This we will discuss a bit later.

Example 1 : Multiply a monomial 5a with a binomial  6a²+ b. Verify your result by substituting a = 2 and b = 3

5a * (6a²+ b) =  5a * 6a²   +  5a b

                        = 30a³ + 5ab

LHS : (for a = 2, b= 3)

      =  5 * 2 ( 6 * 4 + 3)

    =    10 ( 27)   = 270

RHS : (for a = 2, b= 3)

            = 30 * 2³  +  5 * 2 * 3

        =   240   +  30

          = 270

Thus multiplication of  a monomial 5a with a binomial  6a²+ b is 30a³+  5a b.

Example 2 : Multiply  x² ñ y²   with 4xy. Verify your results with x = 11, y = 3

(x² ñ y² ) *  4xy =  4x²xy   - y²4xy

                     = 4x³y ñ 4xy³

LHS : ( for x = 11, y = 3)

            = (11*11 ñ 3 * 3) * 4 * 11 * 3

            = ( 121 ñ 9 ) * 132

            = 112 * 132 = 14784

RHS : ( for x = 11, y = 3)

            = 4 * (11)³ * 3 ñ 4 * 11 * (3) ³

            = 4 * 1331 * 3 ñ 44 * 27

            = 15972 ñ 1188

            = 14784

Thus LHS = RHS  and multiplication of  x² ñ y²   with 4xy is 4x³y ñ 4xy³.

Example 3 : Multiply two binomials (2x-y) and (x+4y)

(2x-y) * (x+4y) = (2x) * (x + 4y)  ñ y * (x+4y)

                                =  2x² + 8xy  ñ yx ñ 4xy

                                =  2x² + 3xy                              ( yx = xy)

You can see in this example how coefficients of same variables are clubbed together. Here (8xy  ñ yx ñ 4xy) are added together to give 3 xy.

You can put any value of x and y and you will see that LHS = RHS.

3. Polynomials
Mathematical operations with monomials and binomials give rise to polynomials.

In the Example 3 above, if add a constant to 2x² + 3xy, then the algebraic expression becomes a trinomial. 2x² + 3xy + 5 is a trinomial as it has three terms, none of the terms have anything common in them. x + y + 7 is also a trinomial. Other examples of trinomials are x + y + z, x² y + z + xy.

If an algebraic expression has more than three independent terms, then it is called a polynomial. For example : 9 + x + x² + x³ is called a polynomial with four terms.

Now we can understand why algebraic expressions such as 1/x or z/ y² are not monomials.

Let     1      = 10
      
         x

10x = 1

10x ñ 1 = 0 is a binomial  and not a monomial!!

Similarly  let   z    = 3
                  
                     y²

z = 3y²

z - 3y²   =  0 is again a binomial and not a monomial!!  

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