%5E3-expansion.jpeg "(a+b+c+d)^3 Expansion")
(a+b+c+d)^3 Expansion: A Detailed Guide
Expanding powers of sums of variables can be a daunting task, especially when dealing with higher powers. In this article, we will dive into the expansion of (a+b+c+d)^3, a common expression in algebra and mathematics.
What is the Formula for Expanding (a+b+c+d)^3?
The formula for expanding (a+b+c+d)^3 is based on the principle of binomial expansion. The general formula for expanding (a+b)^n is:
(a+b)^n = a^n + na^(n-1)b + n(n-1)a^(n-2)b^2 + ... + b^n
To expand (a+b+c+d)^3, we can apply this formula twice, first treating a+b+c as a single variable, and then expanding the resulting expression further.
Step-by-Step Expansion of (a+b+c+d)^3
Step 1: Expand (a+b+c)^3
Using the formula, we get:
(a+b+c)^3 = a^3 + 3a^2(b+c) + 3a(b+c)^2 + (b+c)^3
Step 2: Expand (b+c+d)^3
Similarly, we get:
(b+c+d)^3 = b^3 + 3b^2(c+d) + 3b(c+d)^2 + (c+d)^3
Step 3: Combine and Simplify
Now, we combine the two expressions and simplify:
(a+b+c+d)^3 = a^3 + 3a^2(b+c+d) + 3a((b+c)^2 + 2(b+c)d + d^2) + ((b+c+d)^3)
Step 4: Expand and Simplify Further
Expanding and simplifying the expression, we get:
(a+b+c+d)^3 = a^3 + 3a^2b + 3a^2c + 3a^2d + 6abc + 6abd + 6acd + 3bcd + b^3 + 3b^2c + 3b^2d + c^3 + 3c^2d + d^3
The Final Result
The final expansion of (a+b+c+d)^3 is:
(a+b+c+d)^3 = a^3 + 3a^2b + 3a^2c + 3a^2d + 6abc + 6abd + 6acd + 3bcd + b^3 + 3b^2c + 3b^2d + c^3 + 3c^2d + d^3
This expansion can be useful in various mathematical and algebraic applications, such as solving equations, finding derivatives, and more.
Conclusion
Expanding (a+b+c+d)^3 may seem like a complex task, but by applying the principle of binomial expansion and following the step-by-step process, we can arrive at the final result. This expansion is a valuable tool in algebra and mathematics, and understanding its derivation can help build a stronger foundation in these subjects.
