%5E3-all-formula.jpeg "(a+b+c)^3 All Formula")
The Formula for (a+b+c)^3
The formula for (a+b+c)^3 is a well-known result in algebra, and it's a great example of how the binomial theorem can be extended to three or more variables. In this article, we'll derive the formula and provide some examples to illustrate its use.
The Binomial Theorem
Before we dive into the formula for (a+b+c)^3, let's quickly review the binomial theorem for two variables:
$(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$
where $\binom{n}{k}$ is the binomial coefficient, also written as $n \choose k$.
Deriving the Formula
To find the formula for (a+b+c)^3, we can start by expanding the expression using the distributive property:
$(a+b+c)^3 = (a+b+c)(a+b+c)(a+b+c)$
Now, let's multiply out the first two factors:
$(a+b+c)(a+b+c) = a^2 + 2ab + b^2 + 2ac + 2bc + c^2$
Next, we multiply the result by (a+b+c) again:
$(a^2 + 2ab + b^2 + 2ac + 2bc + c^2)(a+b+c)$
After multiplying out the terms, we get:
$a^3 + 3a^2b + 3ab^2 + b^3 + 3a^2c + 6abc + 3b^2c + 3ac^2 + 3bc^2 + c^3$
The Final Formula
After collecting like terms, we arrive at the final formula:
$(a+b+c)^3 = a^3 + 3a^2b + 3ab^2 + b^3 + 3a^2c + 6abc + 3b^2c + 3ac^2 + 3bc^2 + c^3$
Examples
Let's try out the formula with some examples:
Example 1
Find the value of (2+3+4)^3:
$(2+3+4)^3 = 2^3 + 3\cdot 2^2 \cdot 3 + 3 \cdot 2 \cdot 3^2 + 3^3 + 3\cdot 2^2 \cdot 4 + 6 \cdot 2 \cdot 3 \cdot 4 + 3 \cdot 3^2 \cdot 4 + 3 \cdot 2 \cdot 4^2 + 3 \cdot 3 \cdot 4^2 + 4^3$
After evaluating the expression, we get:
$(2+3+4)^3 = 125$
Example 2
Find the value of (x+2y+z)^3:
$(x+2y+z)^3 = x^3 + 3x^2(2y) + 3x(2y)^2 + (2y)^3 + 3x^2z + 6x(2y)z + 3(2y)^2z + 3xz^2 + 3(2y)z^2 + z^3$
After simplifying the expression, we get:
$(x+2y+z)^3 = x^3 + 6x^2y + 12xy^2 + 8y^3 + 3x^2z + 12xyz + 12y^2z + 3xz^2 + 6yz^2 + z^3$
I hope this article has helped you understand the formula for (a+b+c)^3 and how to apply it to different problems. Happy calculating!
