%5E2.jpeg "(a+b+c+d+e+f+g+h+i)^2")
Expanding (a+b+c+d+e+f+g+h+i)^2
The expression (a+b+c+d+e+f+g+h+i)^2 represents the square of the sum of nine variables. Expanding this expression can be quite tedious, but there's a systematic way to do it.
Understanding the Concept
The expression (a+b+c+d+e+f+g+h+i)^2 is essentially multiplying the sum of the variables by itself:
(a+b+c+d+e+f+g+h+i) * (a+b+c+d+e+f+g+h+i)
To expand this, we need to distribute each term in the first set of parentheses to every term in the second set.
Using the FOIL Method (Generalized)
While the FOIL method is commonly used for expanding binomials, we can generalize it to handle multiple variables. The process involves:
- First: Multiply the first terms of each set of parentheses.
- Outer: Multiply the outer terms of each set of parentheses.
- Inner: Multiply the inner terms of each set of parentheses.
- Last: Multiply the last terms of each set of parentheses.
However, with nine variables, this will result in a lot of terms!
The Resulting Expression
After expanding the entire expression, we get a sum of squares and cross-product terms:
(a+b+c+d+e+f+g+h+i)^2 =
- a² + b² + c² + d² + e² + f² + g² + h² + i²
- + 2ab + 2ac + 2ad + 2ae + 2af + 2ag + 2ah + 2ai
- + 2bc + 2bd + 2be + 2bf + 2bg + 2bh + 2bi
- + 2cd + 2ce + 2cf + 2cg + 2ch + 2ci
- + 2de + 2df + 2dg + 2dh + 2di
- + 2ef + 2eg + 2eh + 2ei
- + 2fg + 2fh + 2fi
- + 2gh + 2gi
- + 2hi
This expanded expression consists of:
- Nine squared terms: Each variable multiplied by itself.
- 36 cross-product terms: Each unique combination of two distinct variables, multiplied by 2.
Key Points:
- The number of terms in the expanded expression grows rapidly with the number of variables.
- This expansion can be used to simplify expressions involving squared sums.
- It's crucial to be organized and methodical when expanding such expressions to avoid errors.