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Use the Binomial Theorem to complete the expansion of $(x + 2)^4$ . $\newline$ $x^4 + 8x^3 + 24x^2 + 32x + \underline{\hspace{1cm}}$

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Binomial Theorem II

Full solution

Q. Use the Binomial Theorem to complete the expansion of

(x + 2)^4

.

\newline

x^4 + 8x^3 + 24x^2 + 32x + \underline{\hspace{1cm}}

Identify values of a, b, n: Identify the values of a, b, n from the expression $(x + 2)^4$ . Here, $a = x$ , $b = 2$ , and $n = 4$ .
Recognize pattern of binomial expansion: Recognize the pattern of the binomial expansion to determine the index of the missing term. The given expansion is $x^4 + 8x^3 + 24x^2 + 32x + \_\_\_\_$ . The missing term is the constant term, which corresponds to $r = 4$ in the binomial expansion.
Use binomial coefficient formula: Use the binomial coefficient formula to find the missing term. The general term in a binomial expansion is given by $\binom{n}{r} \cdot a^{n-r} \cdot b^r$ . For the missing term, we have $\binom{4}{4} \cdot x^{4-4} \cdot 2^4$ .
Calculate binomial coefficient: Calculate the binomial coefficient $\binom{4}{4}$ , which is equal to $1$ because any number choose itself is $1$ .
Simplify expression for missing term: Simplify the expression for the missing term. We have $1 \times x^{(4-4)} \times 2^4$ , which simplifies to $1 \times x^0 \times 16$ , since $x^0$ is $1$ .
Calculate value of missing term: Calculate the value of the missing term. The expression simplifies to $1 \times 1 \times 16$ , which equals $16$ .

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