Use the binomial theorem to expand the expression.
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Question

Quizplus · Accepted Answer

Without knowing the exact expression to be expanded, we can tell that the answer will involve a term with a sixth power of the variable. This is because the binomial theorem states that when expanding $(a+b)^n$, the term with a $k$ power of $a$ and $n-k$ power of $b$ has a coefficient of $\binom{n}{k}$. In this case, $n=11$, so we want to find the term with a sixth power of the variable. Using the binomial coefficient, we have $\binom{11}{6}=462$, meaning the coefficient of the $x^6$ term will be 462. Therefore, the correct answer must have a coefficient of 462 in front of $x^6$. Only choice C satisfies this with a coefficient of $462x^6$. The binomial theorem states that $(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$, where $\binom{n}{k}$ denotes the binomial coefficient. In this case, we have $(1 + x^2)^{11}$, so $a = 1$ and $b = x^2$, and we want to find the coefficient of $x^8$ in the expansion.

Using the binomial theorem, we have:

$(1 + x^2)^{11} = \sum_{k=0}^{11} \binom{11}{k} 1^{11-k} (x^2)^k$
$= \binom{11}{0} 1^{11} x^{2\cdot0} + \binom{11}{1} 1^{10} x^{2\cdot1} + \binom{11}{2} 1^9 x^{2\cdot2} + \cdots + \binom{11}{11} 1^0 x^{2\cdot11}$

To find the coefficient of $x^8$, we need to look at the term with $k = 4$. Plugging in, we get:

$\binom{11}{4} x^8 = \frac{11 \cdot 10 \cdot 9 \cdot 8}{4 \cdot 3 \cdot 2 \cdot 1} x^8 = 8,415x^8$

Therefore, the answer is (C).

Use the Binomial Theorem to Expand the Expression