Παρασκευή, 22 Σεπτεμβρίου 2017

Expanding Trinomials $(a+b+c)^n$

Expanding Trinomials

Observations:
        
Rewrite the right hand side in triangular form:



1a2


     
2ab

2ac

1b2

2bc

1c2

Divide the triangle into variable part and the coefficient part:



a2






1



ab

ac




2

2

b2

bc

c2


1

2

1

(1) For the variable part on the left hand side :

Note that for the highest power of  “a” is on the top of the triangle and the powers are in descending order towards the base of the triangle. The highest power of  “b” is in the lower left corner and the powers are in descending order towards the base of the triangle on the upper right. Similarly, the highest power of “c” is in the lower right corner and the powers are in descending order towards the base of the triangle on the upper left.

(2) For the coefficient part on the right hand side :

It seems easy to remember the coefficient triangle. But are there any pattern? Let us increase the power of the trinomial.
        


More observations:

         

The variable triangle and the coefficient triangle are as follows:




a3







1





a2b
a2c




3

3



ab2
abc
ac2


3

6

3

b3

b2c
bc2
c3

1

3

3

1

(1) Do you know how to make variable triangle? Study carefully the powers of a, b, c.

(2) The coefficient triangle on the right hand side can be made as follows:

 
 
 
1
 
 
 


1


 
 
 
1
 
 
 
 
 
1
 
1
 
 

x
3

=
 
 
3
 
3
 
 
 
1
 
2
 
1
 


3


 
3
 
6
 
3
 
1
 
3
 
3
 
1


1


1
 
3
 
3
 
1

The left most is the Pascal triangle. The coefficients are multiplied correspondingly by (1,3,3,1), that is, the last line of the Pascal triangle placing vertically. You can get the coefficient triangle in the trinomial expansion by finding the product.


Exercise :
         Expand
                         .


Hint:
The coefficient triangle is as follows.

 
 
 
 
1
 
 
 
 

1


 
 
 
 
1
 
 
 
 
 
 
 
1
 
1
 
 
 
x
4

=
 
 
 
4
 
4
 
 
 
 
 
1
 
2
 
1
 
 

6


 
 
6
 
12
 
6
 
 
 
1
 
3
 
3
 
1
 

4


 
4
 
12
 
12
 
4
 
1
 
4
 
6
 
4
 
1

1


1
 
4
 
6
 
4
 
1


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