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

Jasonbradley

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Jun 17, 2024

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

The expression (x-y)(x^2 + 2xy + y^2) represents the product of two factors:

• (x-y): This is a simple binomial difference of two terms.
• (x^2 + 2xy + y^2): This is a perfect square trinomial, which can be factored into (x+y)^2.

Understanding the Concept:

The given expression is an example of a pattern that appears frequently in algebra, known as the difference of squares. The difference of squares pattern states:

(a - b)(a + b) = a^2 - b^2

Applying the Pattern:

Let's apply the difference of squares pattern to our expression. We can rewrite the given expression as follows:

1. Recognize the pattern: Notice that (x^2 + 2xy + y^2) is a perfect square trinomial that can be factored into (x+y)^2.
2. Substitute and simplify: (x - y)(x^2 + 2xy + y^2) = (x - y)(x + y)^2

Now, we can apply the difference of squares pattern where:

• a = x
• b = (x+y)

Therefore, we have:

(x - y)(x + y)^2 = x^2 - (x+y)^2

Expanding and Simplifying:

Expanding the square and simplifying, we get:

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

Final Result:

The simplified form of the expression (x-y)(x^2 + 2xy + y^2) is -2xy - y^2.