%5E3%2B(x-y)%5E2%2B1%2F3(x-y)%2B1%2F27.jpeg "(x-y)^3+(x-y)^2+1/3(x-y)+1/27")
Factoring and Simplifying (x-y)^3 + (x-y)^2 + 1/3(x-y) + 1/27
The expression (x-y)^3 + (x-y)^2 + 1/3(x-y) + 1/27 might seem intimidating at first glance, but it can be simplified through factoring and recognizing a pattern. Here's how:
Recognizing the Pattern
Notice that the expression resembles a perfect cube expansion. Recall the formula for the cube of a binomial:
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
Let's make a substitution to see this pattern more clearly. Let:
- a = (x-y)
- b = 1/3
Now, our expression becomes:
a^3 + a^2b + ab^2 + b^3
Factoring the Expression
We can now factor this expression as a perfect cube:
(a + b)^3 = (x-y + 1/3)^3
Simplifying the Expression
Therefore, the simplified form of the original expression is:
(x-y)^3 + (x-y)^2 + 1/3(x-y) + 1/27 = (x-y + 1/3)^3
Conclusion
By recognizing the pattern of a perfect cube expansion and applying the appropriate substitution, we were able to simplify the complex expression to a much more manageable form. This demonstrates the power of factoring and pattern recognition in simplifying algebraic expressions.