To expand the expression \((x + 1)^4\) using the binomial theorem, we can apply the formula:
\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
In this case, \(a = x\), \(b = 1\), and \(n = 4\).
Now we can write the expansion:
\[ (x + 1)^4 = \sum_{k=0}^{4} \binom{4}{k} x^{4-k} (1)^k \]
Calculating each term of the sum:
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For \(k = 0\): \[ \binom{4}{0} x^{4-0} (1)^0 = 1 \cdot x^4 \cdot 1 = x^4 \]
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For \(k = 1\): \[ \binom{4}{1} x^{4-1} (1)^1 = 4 \cdot x^3 \cdot 1 = 4x^3 \]
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For \(k = 2\): \[ \binom{4}{2} x^{4-2} (1)^2 = 6 \cdot x^2 \cdot 1 = 6x^2 \]
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For \(k = 3\): \[ \binom{4}{3} x^{4-3} (1)^3 = 4 \cdot x^1 \cdot 1 = 4x \]
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For \(k = 4\): \[ \binom{4}{4} x^{4-4} (1)^4 = 1 \cdot x^0 \cdot 1 = 1 \]
Next, combine all these terms together:
\[ (x + 1)^4 = x^4 + 4x^3 + 6x^2 + 4x + 1 \]
Thus, the polynomial in standard form is:
\[ \boxed{x^4 + 4x^3 + 6x^2 + 4x + 1} \]
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