%5E6-expansion.jpeg "(e-1)^6 Expansion")
Binomial Expansion: (e-1)^6
In mathematics, the binomial theorem is a powerful tool for expanding powers of a binomial expression, such as (e-1)^6. In this article, we will explore the expansion of (e-1)^6 using the binomial theorem.
The Binomial Theorem
The binomial theorem states that for any positive integer n and real numbers a and b:
$(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$
where $\binom{n}{k}$ is the binomial coefficient, given by:
$\binom{n}{k} = \frac{n!}{k!(n-k)!}$
Expanding (e-1)^6
To expand (e-1)^6, we can use the binomial theorem with a = e and b = -1. Substituting these values into the theorem, we get:
$(e-1)^6 = \sum_{k=0}^6 \binom{6}{k} e^{6-k} (-1)^k$
Now, let's compute the binomial coefficients:
$\binom{6}{0} = \frac{6!}{0!(6-0)!} = 1$ $\binom{6}{1} = \frac{6!}{1!(6-1)!} = 6$ $\binom{6}{2} = \frac{6!}{2!(6-2)!} = 15$ $\binom{6}{3} = \frac{6!}{3!(6-3)!} = 20$ $\binom{6}{4} = \frac{6!}{4!(6-4)!} = 15$ $\binom{6}{5} = \frac{6!}{5!(6-5)!} = 6$ $\binom{6}{6} = \frac{6!}{6!(6-6)!} = 1$
Substituting these coefficients into the expansion, we get:
$(e-1)^6 = e^6 - 6e^5 + 15e^4 - 20e^3 + 15e^2 - 6e + 1$
Simplifying the Expansion
We can simplify the expansion by combining like terms:
$(e-1)^6 = e^6 - 6e^5 + 15e^4 - 20e^3 + 15e^2 - 6e + 1$
Conclusion
In this article, we have seen how to expand (e-1)^6 using the binomial theorem. The expansion is a polynomial of degree 6, with coefficients that can be computed using the binomial theorem. This expansion has many applications in mathematics, physics, and engineering.
