If you expand the polynomial , the resulting polynomial has three terms whose coefficients are 1, 2, and 1. Find the sum of the coefficients of the terms in the expansion of .
Source: mathcontest.olemiss.edu 5/5/2008
SOLUTION
We use the “area” model to represent the product of polynomials. For example, is thought of as where and .
The product is calculated as follows:

a 
b 

a 
a² 
ab 

b 
ab 
b² 
Thus, .
Sum of the coefficients: .
The problem asks for the sum of the coefficients, not what the individual coefficients look like in the finished product so it is not necessary to combine the like terms before calculating the sum of the coefficients. Instead, we will leave all the like terms uncombined as follows:
. The sum of the coefficients is
Similarly, the product is calculated as follows:

a² 
2ab 
b² 

a 
a³ 
2a²b 
ab² 

b 
a²b 
2ab² 
b³ 
Thus, .
Sum of the coefficients: .
If we left the like terms uncombined, the same operation looks like this:

a² 
ab 
ab 
b² 

a 
a³ 
a²b 
a²b 
ab² 

b 
a²b 
ab² 
ab² 
b³ 
All 8 coefficients are equal to 1. Their sum is .
Better yet, let’s leave out all the variables and just use the coefficients. The same operation looks like this:

1 
1 
1 
1 

1 
1 
1 
1 
1 

1 
1 
1 
1 
1 
Let’s use this procedure and calculate the sum of the coefficients of .

1 
1 
1 
1 
1 
1 
1 
1 

1 
1 
1 
1 
1 
1 
1 
1 
1 

1 
1 
1 
1 
1 
1 
1 
1 
1 
The sum of the coefficients equals 16.
For verification, let’s calculate the product the old way and see if we get the same result:

a³ 
3a²b 
3ab² 
b³ 

a 
a⁴ 
3a³b 
3a²b² 
ab³ 

b 
a³b 
3a²b² 
3ab³ 
b⁴ 
Thus, .
Sum of the coefficients: .
General formula
Let be the sum of the coefficients of . Then,
.
.
.
In general, for :
.
Sum of coefficients of .
.
If you are familiar with the Pascal Triangle, you can derive the individual coefficients of and calculate their sum as follows:
.
Answer: 1024.
Right on!