Question

express the following polynomial in expanded form and simplify completely.
\\p(x) = (x + 4)^4\\
\\p(x) =\\

Explanation:

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<reasoning>

Apply the Binomial Theorem for exponent 4

\[
(a + b)^4 = \binom{4}{0}a^4 + \binom{4}{1}a^3b + \binom{4}{2}a^2b^2 + \binom{4}{3}ab^3 + \binom{4}{4}b^4
\]
\[
\binom{4}{0}=1, \quad \binom{4}{1}=4, \quad \binom{4}{2}=6, \quad \binom{4}{3}=4, \quad \binom{4}{4}=1
\]

Substitute x and 4 into the expansion

\[

$$\begin{aligned} p(x) &= 1(x)^4 + 4(x)^3(4)^1 + 6(x)^2(4)^2 + 4(x)^1(4)^3 + 1(4)^4 \\ p(x) &= x^4 + 4 \cdot 4 x^3 + 6 \cdot 16 x^2 + 4 \cdot 64 x + 256 \end{aligned}$$

\]

Simplify the coefficients

\[
p(x) = x^4 + 16x^3 + 96x^2 + 256x + 256
\]
</reasoning>

<answer>
Express the following polynomial in expanded form and simplify completely.
\(p(x) = (x + 4)^4\)

\(p(x) =\) <blank>\(x^4 + 16x^3 + 96x^2 + 256x + 256\)</blank>
</answer>

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</post_analysis>

Answer:

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</pre_analysis>

<reasoning>

Apply the Binomial Theorem for exponent 4

\[
(a + b)^4 = \binom{4}{0}a^4 + \binom{4}{1}a^3b + \binom{4}{2}a^2b^2 + \binom{4}{3}ab^3 + \binom{4}{4}b^4
\]
\[
\binom{4}{0}=1, \quad \binom{4}{1}=4, \quad \binom{4}{2}=6, \quad \binom{4}{3}=4, \quad \binom{4}{4}=1
\]

Substitute x and 4 into the expansion

\[

$$\begin{aligned} p(x) &= 1(x)^4 + 4(x)^3(4)^1 + 6(x)^2(4)^2 + 4(x)^1(4)^3 + 1(4)^4 \\ p(x) &= x^4 + 4 \cdot 4 x^3 + 6 \cdot 16 x^2 + 4 \cdot 64 x + 256 \end{aligned}$$

\]

Simplify the coefficients

\[
p(x) = x^4 + 16x^3 + 96x^2 + 256x + 256
\]
</reasoning>

<answer>
Express the following polynomial in expanded form and simplify completely.
\(p(x) = (x + 4)^4\)

\(p(x) =\) <blank>\(x^4 + 16x^3 + 96x^2 + 256x + 256\)</blank>
</answer>

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