%5E2-expansion.jpeg "(a+b+c+d)^2 Expansion")
Expanding the Square of a Sum: (a+b+c+d)^2
Expanding the square of a sum, like (a+b+c+d)^2, can seem daunting at first, but it follows a simple pattern. This pattern involves multiplying each term in the sum by itself and by every other term, and then summing all the results.
Here's a breakdown of the process:
Understanding the Pattern
The general form for expanding (a+b+c+d)^2 is:
(a + b + c + d)^2 = a^2 + b^2 + c^2 + d^2 + 2ab + 2ac + 2ad + 2bc + 2bd + 2cd
Notice the following:
- Squares of individual terms: Each variable is squared (a^2, b^2, c^2, d^2).
- Cross-products: Every possible combination of two different variables is multiplied and then doubled (2ab, 2ac, 2ad, 2bc, 2bd, 2cd).
Visualizing the Expansion
You can visualize the expansion using a table:
| a | b | c | d | |
|---|---|---|---|---|
| a | a^2 | ab | ac | ad |
| b | ab | b^2 | bc | bd |
| c | ac | bc | c^2 | cd |
| d | ad | bd | cd | d^2 |
Each cell represents the product of the corresponding row and column elements. The diagonal elements represent the squares of individual terms, and the off-diagonal elements represent the cross-products.
Example
Let's expand (x + 2y + 3z)^2 using the formula:
(x + 2y + 3z)^2 = x^2 + (2y)^2 + (3z)^2 + 2(x)(2y) + 2(x)(3z) + 2(2y)(3z)
Simplifying:
(x + 2y + 3z)^2 = x^2 + 4y^2 + 9z^2 + 4xy + 6xz + 12yz
Key Points to Remember
- The pattern remains the same regardless of the number of terms in the sum.
- The expansion results in a polynomial with a degree of 2 (quadratic).
- You can use the table method for visualization, especially when dealing with longer sums.
By understanding this pattern, you can efficiently expand squares of sums containing any number of terms.