(3x+2y+2z)(9x^2+4y^2+4z^2-6xy-4yz-6zx)

2 min read Jun 16, 2024
(3x+2y+2z)(9x^2+4y^2+4z^2-6xy-4yz-6zx)

Expanding the Expression (3x+2y+2z)(9x^2+4y^2+4z^2-6xy-4yz-6zx)

This expression is in the form of a sum of cubes. We can use the following formula to expand it:

(a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) = a^3 + b^3 + c^3 - 3abc

In this case, we have:

  • a = 3x
  • b = 2y
  • c = 2z

Let's apply the formula:

(3x + 2y + 2z)(9x^2 + 4y^2 + 4z^2 - 6xy - 4yz - 6zx) = (3x)^3 + (2y)^3 + (2z)^3 - 3(3x)(2y)(2z)

Now, let's simplify the expression:

(3x)^3 + (2y)^3 + (2z)^3 - 3(3x)(2y)(2z) = 27x^3 + 8y^3 + 8z^3 - 36xyz

Therefore, the expanded form of the given expression is 27x^3 + 8y^3 + 8z^3 - 36xyz.

Key Points:

  • Recognizing the expression as a sum of cubes allows for quick and efficient expansion.
  • The formula helps avoid lengthy and error-prone multiplications.
  • Understanding this type of expression is useful in various mathematical applications, including algebra and calculus.

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