Expand the binomial by using the binomial theorem.
-$$\left( e ^ { x } - e ^ { - x } \right) ^ { 4 }$$
A) $e ^ { 4 x } + 4 e ^ { 3 x } + 6 e ^ { 2 x } + 4 e ^ { x } + 1$
B) $e ^ { 4 x } + 4 e ^ { 2 x } + 6 + \frac { 4 } { e ^ { 2 x } } + \frac { 1 } { e ^ { 4 x } }$
C) $e ^ { 4 x } - 4 e ^ { 3 x } + 6 e ^ { 2 x } - 4 e ^ { x } + 1$
D) $e ^ { 4 x } - 4 e ^ { 2 x } + 6 - \frac { 4 } { e ^ { 2 x } } + \frac { 1 } { e ^ { 4 x } }$

Question

Quizplus · Accepted Answer

The binomial theorem states that $(a - b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}(-b)^k$. Applying this to $(e^x - e^{-x})^4$, we get terms with alternating signs and powers of $e^x$ and $e^{-x}$, leading to the correct expansion in option D.

Expand the Binomial by Using the Binomial Theorem (ex−e−x)4\left( e ^ { x } - e ^ { - x } \right) ^ { 4 }(ex−e−x)4

Expand the Binomial by Using the Binomial Theorem $\left( e ^ { x } - e ^ { - x } \right) ^ { 4 }$