(x+2y)(x^2-2xy+4y^2)+(2x-y)(4x^2+2xy+y^2)

Jasonbradley

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Jun 16, 2024

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Simplifying the Expression: (x+2y)(x^2-2xy+4y^2)+(2x-y)(4x^2+2xy+y^2)

This expression involves the multiplication of two sets of binomials. To simplify it, we can use the following algebraic identities:

1. Sum of Cubes: a³ + b³ = (a + b)(a² - ab + b²)

2. Difference of Cubes: a³ - b³ = (a - b)(a² + ab + b²)

Let's break down the simplification process:

Step 1: Identify the patterns

Notice that in the first set of parentheses, we have the pattern of the sum of cubes (x³ + (2y)³) and in the second set, we have the difference of cubes ((2x)³ - y³).

Step 2: Apply the identities

Using the identities mentioned above, we can rewrite the expression as follows:

(x+2y)(x^2-2xy+4y^2) = x³ + (2y)³

(2x-y)(4x^2+2xy+y^2) = (2x)³ - y³

Step 3: Combine and simplify

Now, our expression becomes:

x³ + (2y)³ + (2x)³ - y³

Simplifying further, we get:

9x³ + 7y³

Therefore, the simplified form of the expression (x+2y)(x^2-2xy+4y^2)+(2x-y)(4x^2+2xy+y^2) is 9x³ + 7y³.