Use the binomial theorem to expand the expression.
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Question

Quizplus · Accepted Answer

The binomial theorem states that $(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^{k}$ where $\binom{n}{k}$ represents the number of ways to choose $k$ items from a set of $n$ items. In this case, we have $(1+eb)^{4}$, so $a = 1$ and $b = eb$, and $n=4$. Plugging this into the formula, we get:

$$(1+eb)^4 = \binom{4}{0}1^4(eb)^0 + \binom{4}{1}1^3(eb)^1 + \binom{4}{2}1^2(eb)^2 + \binom{4}{3}1^1(eb)^3 + \binom{4}{4}1^0(eb)^4$$

Simplifying, we have:

$$(1+eb)^4 = eb^4 + 4eb^3 + 6eb^2 + 4eb + 1$$

Therefore, the correct choice is C. The binomial theorem states that the expansion of $(a+b)^n$ can be written as $\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k$. In this case, $a=1$ and $b=e^x$. So we have $(1+e^x)^{11}=\sum_{k=0}^{11}\binom{11}{k}(1)^{11-k}(e^x)^k$. Simplifying the expression, we get the term with the highest power of $x$ to be $\binom{11}{4}e^{4x}$, which matches option C.

Use the Binomial Theorem to Expand the Expression