%5E3-formula-expansion.jpeg "(a+b+c)^3 Formula Expansion")
Formula Expansion of (a+b+c)^3
In algebra, expanding the cube of a trinomial expression, such as (a+b+c)^3, can be a daunting task. However, with the right approach, it can be simplified and expanded into a more manageable form.
Step 1: Apply the Binomial Theorem
To expand (a+b+c)^3, we can use the Binomial Theorem, which states that:
$(x+y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k$
where $n$ is a positive integer, and $\binom{n}{k}$ is the binomial coefficient.
Step 2: Adapt the Binomial Theorem to (a+b+c)^3
Since we have a trinomial expression, (a+b+c), we need to adapt the Binomial Theorem to accommodate three variables. We can do this by applying the theorem twice:
$(a+b+c)^3 = (a+(b+c))^3 = \sum_{k=0}^3 \binom{3}{k} a^{3-k} (b+c)^k$
Step 3: Expand the Inner Binomial
Now, we need to expand the inner binomial $(b+c)^k$:
$(b+c)^k = \sum_{j=0}^k \binom{k}{j} b^{k-j} c^j$
Step 4: Combine and Simplify
Substitute the expanded inner binomial back into the original expression and simplify:
$(a+b+c)^3 = \sum_{k=0}^3 \binom{3}{k} a^{3-k} \sum_{j=0}^k \binom{k}{j} b^{k-j} c^j$
After expanding and simplifying, we get:
$(a+b+c)^3 = a^3 + 3a^2b + 3a^2c + 3ab^2 + 6abc + 3ac^2 + b^3 + 3b^2c + 3bc^2 + c^3$
Conclusion
In conclusion, the formula expansion of (a+b+c)^3 is a complex process that involves applying the Binomial Theorem twice and simplifying the resulting expression. With practice and patience, you can master this process and expand other higher-degree trinomial expressions with ease.
