(x%5E2%2By%5E2)(x-y)%5E2(x%2By).jpeg "(x^2+xy+y^2)(x^2+y^2)(x-y)^2(x+y)")
Factoring and Simplifying (x^2+xy+y^2)(x^2+y^2)(x-y)^2(x+y)
This expression is a combination of several factors. Let's break it down and simplify it:
Understanding the Factors
- (x^2 + xy + y^2): This is a quadratic expression that cannot be factored further using real numbers. It's often called the "sum of squares with a cross-term."
- (x^2 + y^2): This is also a quadratic expression representing the sum of squares. It cannot be factored further using real numbers.
- (x - y)^2: This is the square of the difference of two terms, which expands to (x - y)(x - y) = x^2 - 2xy + y^2.
- (x + y): This is the sum of two terms.
Simplifying the Expression
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Expand the squares:
- (x - y)^2 = x^2 - 2xy + y^2
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Rearrange the factors:
- (x^2 + xy + y^2)(x^2 + y^2)(x^2 - 2xy + y^2)(x + y)
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Combine terms: At this point, we can't simplify further without specific values for x and y. The expression is in its most simplified form, as it is a product of irreducible factors.
Key Points
- This expression can't be factored further with real numbers due to the presence of the sum of squares terms (x^2 + xy + y^2) and (x^2 + y^2).
- The simplified form is a product of four factors, each with a specific algebraic structure.
This exercise highlights the importance of recognizing different algebraic forms and their properties when simplifying complex expressions.