%5E3-formula-proof.jpeg "(x+y+z)^3 Formula Proof")
The Formula Proof of (x+y+z)^3
In algebra, the expansion of $(x+y+z)^3$ is a fundamental concept that is widely used in various mathematical operations. In this article, we will delve into the formula proof of $(x+y+z)^3$ and explore its significance in mathematics.
The Formula:
The formula for $(x+y+z)^3$ is given by:
$(x+y+z)^3 = x^3 + y^3 + z^3 + 3xy(x+y) + 3xz(x+z) + 3yz(y+z)$
Proof:
To prove the formula, we can start by using the binomial theorem, which states that:
$(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^k b^{n-k}$
where $n$ is a non-negative integer, and $\binom{n}{k}$ is the binomial coefficient.
Now, let's consider the expression $(x+y+z)^3$. We can write it as:
$(x+y+z)^3 = ((x+y)+z)^3$
Using the binomial theorem, we can expand the inner parentheses:
$(x+y+z)^3 = ((x+y)+z)^3 = \sum_{k=0}^3 \binom{3}{k} (x+y)^k z^{3-k}$
Now, let's compute each term in the sum:
- When $k=0$, we have $\binom{3}{0} (x+y)^0 z^3 = z^3$
- When $k=1$, we have $\binom{3}{1} (x+y)^1 z^2 = 3(x+y)z^2$
- When $k=2$, we have $\binom{3}{2} (x+y)^2 z^1 = 3(x^2 + 2xy + y^2)z$
- When $k=3$, we have $\binom{3}{3} (x+y)^3 z^0 = x^3 + 3x^2y + 3xy^2 + y^3$
Adding up all the terms, we get:
$(x+y+z)^3 = z^3 + 3(x+y)z^2 + 3(x^2 + 2xy + y^2)z + x^3 + 3x^2y + 3xy^2 + y^3$
Now, let's combine like terms:
- The $x^3$ and $y^3$ terms remain unchanged
- The $z^3$ term remains unchanged
- The $x^2y$ and $xy^2$ terms combine to form $3xy(x+y)$
- The $xz^2$ and $yz^2$ terms combine to form $3(x+y)z^2$
- The $x^2z$ and $y^2z$ terms combine to form $3(x^2 + y^2)z$
- The $xyz$ term is absorbed into the $3xy(x+y)$ term
Simplifying the expression, we finally get:
$(x+y+z)^3 = x^3 + y^3 + z^3 + 3xy(x+y) + 3xz(x+z) + 3yz(y+z)$
Conclusion:
In this article, we have successfully derived the formula for $(x+y+z)^3$ using the binomial theorem. The proof involves expanding the expression using the binomial theorem and combining like terms to simplify the result. The formula is widely used in various mathematical operations and is an essential tool for algebraic manipulations.
