(x+y+z)^2x+y/2+z/3)^2-(x/2+y/3+z/4)^2

4 min read Jun 17, 2024

Simplifying the Expression: (x+y+z)^2(x+y/2+z/3)^2-(x/2+y/3+z/4)^2

This expression appears complex, but we can simplify it using algebraic techniques. Here's a breakdown of the steps involved:

1. Recognizing the Pattern

The expression features a difference of squares pattern: (a^2 - b^2) which factors into (a+b)(a-b). Let's define our a and b terms:

a: (x+y+z)(x+y/2+z/3)
b: (x/2+y/3+z/4)

2. Applying the Difference of Squares Formula

Applying the formula, we get:

[(x+y+z)(x+y/2+z/3) + (x/2+y/3+z/4)] * [(x+y+z)(x+y/2+z/3) - (x/2+y/3+z/4)]

3. Expanding the Expressions

Now, we need to expand the expressions inside the brackets. This involves multiplying each term in the first bracket with each term in the second bracket.

Expanding the first bracket:

(x+y+z)(x+y/2+z/3) + (x/2+y/3+z/4) =
x^2 + xy + xz + xy/2 + y^2/2 + yz/2 + xz/3 + yz/6 + z^2/3 + x/2 + y/3 + z/4

Expanding the second bracket:

(x+y+z)(x+y/2+z/3) - (x/2+y/3+z/4) =
x^2 + xy + xz + xy/2 + y^2/2 + yz/2 + xz/3 + yz/6 + z^2/3 - x/2 - y/3 - z/4

4. Combining Like Terms

Finally, we combine like terms in both expanded expressions:

First Bracket:

(1.5)x^2 + (1.5)xy + (1.33)xz + (0.5)y^2 + (0.83)yz + (0.33)z^2 + (0.5)x + (0.33)y + (0.25)z

Second Bracket:

(1.5)x^2 + (1.5)xy + (1.33)xz + (0.5)y^2 + (0.83)yz + (0.33)z^2 - (0.5)x - (0.33)y - (0.25)z

5. The Simplified Expression

The simplified form of the original expression is the product of these two expanded expressions. While it may look lengthy, it is now in a form that is easier to work with for further calculations or analysis.

Important Note: This expression can be further simplified by factoring out common terms and combining constants. The level of simplification depends on the specific application and desired outcome.