%5E6-formula.jpeg "(x-y)^6 Formula")
The (x-y)^6 Formula: Unveiling the Secret to Expanding Binomials
In algebra, expanding binomials is a crucial concept that can be a bit tricky, especially when the exponent is a large number. One such case is the (x-y)^6 formula, which can be daunting for many students. Fear not, dear readers, for we're about to demystify this formula and provide you with a step-by-step guide on how to expand it.
What is the (x-y)^6 Formula?
The (x-y)^6 formula is an extension of the binomial theorem, which states that for any positive integer n, the nth power of a binomial (x+y) can be expanded as:
$(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$
where $\binom{n}{k}$ is the binomial coefficient, which can be calculated using the formula:
$\binom{n}{k} = \frac{n!}{k!(n-k)!}$
Expanding the (x-y)^6 Formula
Now, let's apply the binomial theorem to expand the (x-y)^6 formula. We'll use the formula above, with n=6.
$(x-y)^6 = \sum_{k=0}^{6} \binom{6}{k} x^{6-k} (-y)^k$
Step-by-Step Expansion
Here's the step-by-step expansion of the (x-y)^6 formula:
- k=0: $\binom{6}{0} x^6 (-y)^0 = 1 \cdot x^6 \cdot 1 = x^6$
- k=1: $\binom{6}{1} x^5 (-y)^1 = 6 \cdot x^5 \cdot (-y) = -6x^5y$
- k=2: $\binom{6}{2} x^4 (-y)^2 = 15 \cdot x^4 \cdot y^2 = 15x^4y^2$
- k=3: $\binom{6}{3} x^3 (-y)^3 = 20 \cdot x^3 \cdot (-y)^3 = -20x^3y^3$
- k=4: $\binom{6}{4} x^2 (-y)^4 = 15 \cdot x^2 \cdot y^4 = 15x^2y^4$
- k=5: $\binom{6}{5} x^1 (-y)^5 = 6 \cdot x \cdot (-y)^5 = -6xy^5$
- k=6: $\binom{6}{6} x^0 (-y)^6 = 1 \cdot 1 \cdot (-y)^6 = -y^6$
The Final Answer
Combining all the terms, we get:
$(x-y)^6 = x^6 - 6x^5y + 15x^4y^2 - 20x^3y^3 + 15x^2y^4 - 6xy^5 - y^6$
Voilà! We've successfully expanded the (x-y)^6 formula using the binomial theorem.
Conclusion
Expanding binomials can be a challenging task, but with the binomial theorem, we can tackle even the most daunting formulas, like the (x-y)^6 formula. By following the step-by-step guide, you'll be able to expand this formula with ease and confidence. Remember, practice makes perfect, so be sure to try out other binomial expansions to solidify your understanding of this concept.
