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Expanding the Trinomial Product: (x+3y-z)(2x+5y+3z)(3x-4y-2z)
This article explores the process of expanding the given trinomial product: (x+3y-z)(2x+5y+3z)(3x-4y-2z). We will delve into the steps involved, emphasizing the use of distributive property and the systematic approach to avoid errors.
Step 1: Expanding the First Two Trinomials
Firstly, we focus on expanding the product of the first two trinomials: (x+3y-z)(2x+5y+3z). This can be done using the distributive property, multiplying each term of the first trinomial by every term in the second trinomial.
Expanding (x+3y-z)(2x+5y+3z):
- x(2x+5y+3z) = 2x² + 5xy + 3xz
- 3y(2x+5y+3z) = 6xy + 15y² + 9yz
- -z(2x+5y+3z) = -2xz - 5yz - 3z²
Now, we combine the like terms:
2x² + 5xy + 3xz + 6xy + 15y² + 9yz - 2xz - 5yz - 3z² = 2x² + 11xy + 15y² + xz + 4yz - 3z²
Step 2: Expanding the Result with the Third Trinomial
We now have a simplified trinomial, 2x² + 11xy + 15y² + xz + 4yz - 3z², which needs to be multiplied by the third trinomial, (3x-4y-2z). Again, we apply the distributive property:
Expanding (2x² + 11xy + 15y² + xz + 4yz - 3z²)(3x-4y-2z):
- 2x²(3x-4y-2z) = 6x³ - 8x²y - 4x²z
- 11xy(3x-4y-2z) = 33x²y - 44xy² - 22xyz
- 15y²(3x-4y-2z) = 45xy² - 60y³ - 30y²z
- xz(3x-4y-2z) = 3x²z - 4xyz - 2xz²
- 4yz(3x-4y-2z) = 12xyz - 16y²z - 8yz²
- -3z²(3x-4y-2z) = -9xz² + 12yz² + 6z³
Finally, we combine the like terms:
6x³ - 8x²y - 4x²z + 33x²y - 44xy² - 22xyz + 45xy² - 60y³ - 30y²z + 3x²z - 4xyz - 2xz² + 12xyz - 16y²z - 8yz² - 9xz² + 12yz² + 6z³
= 6x³ + 25x²y + 45xy² - 60y³ - x²z - 14xyz - 48y²z - 11xz² + 4yz² + 6z³
Conclusion
Therefore, the expanded form of the trinomial product (x+3y-z)(2x+5y+3z)(3x-4y-2z) is 6x³ + 25x²y + 45xy² - 60y³ - x²z - 14xyz - 48y²z - 11xz² + 4yz² + 6z³. This step-by-step process demonstrates the methodical approach to expanding complex algebraic expressions.