(9x%5E2%2B4y%5E2%2B4z%5E2-6xy-4yz-6zx).jpeg "(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.