When we evaluate binomial expressions raised to a power, there is a pattern that develops. For example, we take the

binomial expression

**(**

*x*+

*y*) and raise it to the

*n*th power, where

*n*= 1, 2, 3, …...

(

*x*+*y*)1 =*x + y*
(

*x + y*)^2 = (*x + y*)(*x + y*) =*x*^2 +*xy + xy + y^*2 =**2 + 2***x^**xy*+*y*2
(

*x + y*)^3 = (*x + y*)^2(*x + y*) = (*x^*2 + 2*xy*+*y*^2)(*x*+*y*) =*x^*3 +*x*2*y*+ 2*x*^2*y*+ 2*xy*^2 +*y*^2*x*+*y^*3 =**3 + 3***x^**x^*2*y*+ 3*xy*^2 +*y*3
(

*x*+*y*)^4 = (*x + y*)^3(*x*+*y*) = (*x^*3 + 3*x*^2*y*+ 3*xy*^2 +*y^*3)(*x*+*y*) =*x^*4 +*x^*3*y*+ 3*x*^3*y*+ 3*x^*2*y^*2 + 3*x^*2*y^*2 + 3*xy^*3 +*y^*3*x*+*y*4**=**

**4 + 4**

*x^**x^*3

*y*+ 6

*x^*2

*y*^2 + 4

*xy^*3 +

*y*4

Notice that the expansion of each of the binomials is a polynomial. Some patterns observed are as follows:

1. Each expansion begins with the term

*xn*

2. The exponents on

*x*decrease by 1 with each term.
3. The exponents on

*y*begin with 0 (Since*y*0 = 1 there is no*y*in any first term)
4. The exponents on

*y*increase by 1 with each term.
5. The sum of the exponents on any term always equals

*n*.
6. The number of terms is always 1 more than

*n*.
Let's illustrate the preceding 6 patterns with the expansion of (

*x + y*)^3 =*x^*3 + 3*x^*2*y*+ 3*xy^*2 +*y*3
1. The first term is

*x^*3. Since*n*= 3, it holds true that*xn*=*x^*3.
2. The exponents for

*x*on the next terms are 2, 1 and 0, which holds true that the exponents for*x*decrease by 1 each term.
3. There is no

*y*in the first term, therefore it holds true that the exponents for*y*begin with 0 since*y*0 = 1.
4. The exponents for

*y*in the next terms are 1, 2 and 3, which holds true that the exponents for*y*increase by 1 each term.
5. The sum of the exponents per term are as follows: 3 + 0 = 3, 2 + 1 = 3, 1 + 2 = 3 and 0 + 3 = 3. Since

*n*= 3, it holds true that the sum of the exponents of any term equals

*n.*

6. There are 4 terms in this expansion, therefore it holds true that the number of terms is 1 more than

*n*.

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