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Binomial Expansion: (x-y)^6
In algebra, binomial expansion is a method of expanding an expression of the form (a+b)^n, where a and b are variables and n is a positive integer. One of the most common binomial expansions is (x-y)^6, which is a fundamental concept in mathematics.
The Formula
The expansion of (x-y)^6 can be obtained using the binomial theorem, which states that:
$(x-y)^6 = \sum_{k=0}^{6} \binom{6}{k} x^{6-k} (-y)^k$
where $\binom{6}{k}$ is the binomial coefficient, which can be calculated as:
$\binom{6}{k} = \frac{6!}{k!(6-k)!}$
The Expansion
Using the binomial theorem, we can expand (x-y)^6 as follows:
$(x-y)^6 = x^6 - 6x^5y + 15x^4y^2 - 20x^3y^3 + 15x^2y^4 - 6xy^5 + y^6$
Simplification
The above expansion can be simplified by combining like terms:
$(x-y)^6 = x^6 - 6x^5y + 15x^4y^2 - 20x^3y^3 + 15x^2y^4 - 6xy^5 + y^6$
Properties
The expansion of (x-y)^6 has several important properties:
- Symmetry: The expansion is symmetric about the middle term, which is
15x^2y^4. - Alternating signs: The signs of the terms alternate between positive and negative.
- Pascal's Triangle: The coefficients of the terms form a Pascal's triangle, where each coefficient is the sum of the two coefficients directly above it.
Applications
The expansion of (x-y)^6 has numerous applications in various fields, including:
- Algebra: In solving equations and inequalities.
- Geometry: In calculating distances and volumes of geometric shapes.
- Calculus: In finding derivatives and integrals.
- Statistics: In calculating probabilities and statistical distributions.
Conclusion
In conclusion, the expansion of (x-y)^6 is a fundamental concept in algebra and has numerous applications in various fields. Understanding the formula, expansion, and properties of (x-y)^6 is essential for solving problems and advancing in mathematics.
