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Expanding (x+y+z)^3
The expression (x+y+z)^3 represents the cube of the trinomial (x+y+z). Expanding this expression means writing it out in a form where no parentheses remain and all terms are simplified.
There are two common ways to expand (x+y+z)^3:
1. Using the Binomial Theorem
While the Binomial Theorem technically applies to binomials, we can adapt it for trinomials by treating (y+z) as a single unit.
Step 1: Expand using the Binomial Theorem: (x + (y+z))^3 = x^3 + 3x^2(y+z) + 3x(y+z)^2 + (y+z)^3
Step 2: Expand the terms with (y+z) further: x^3 + 3x^2y + 3x^2z + 3xy^2 + 6xyz + 3xz^2 + y^3 + 3y^2z + 3yz^2 + z^3
Step 3: Combine like terms: x^3 + y^3 + z^3 + 3x^2y + 3x^2z + 3xy^2 + 6xyz + 3xz^2 + 3y^2z + 3yz^2
2. Direct Expansion
We can also expand (x+y+z)^3 by multiplying it out directly.
Step 1: Write it out as (x+y+z)(x+y+z)(x+y+z)
Step 2: Multiply the first two binomials: (x^2 + xy + xz + xy + y^2 + yz + xz + yz + z^2)(x+y+z)
Step 3: Simplify the expression: (x^2 + 2xy + 2xz + y^2 + 2yz + z^2)(x+y+z)
Step 4: Multiply the simplified expression by (x+y+z): x^3 + 2x^2y + 2x^2z + xy^2 + 2xyz + xz^2 + x^2y + 2xy^2 + 2xyz + y^3 + 2y^2z + yz^2 + x^2z + 2xyz + 2xz^2 + y^2z + 2yz^2 + z^3
Step 5: Combine like terms: x^3 + y^3 + z^3 + 3x^2y + 3x^2z + 3xy^2 + 6xyz + 3xz^2 + 3y^2z + 3yz^2
Therefore, the expanded form of (x+y+z)^3 is x^3 + y^3 + z^3 + 3x^2y + 3x^2z + 3xy^2 + 6xyz + 3xz^2 + 3y^2z + 3yz^2
This expansion is useful in various mathematical contexts, including algebra, calculus, and physics. It can be used to solve equations, simplify expressions, and understand the relationships between variables.