Question

Expand the expression to a polynomial in standard form:
left parenthesis, x, plus, 1, right parenthesis, to the power 4
(x+1)
4

Answer 1

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:

1. For \(k = 0\):
\[
\binom{4}{0} x^{4-0} (1)^0 = 1 \cdot x^4 \cdot 1 = x^4
\]

2. For \(k = 1\):
\[
\binom{4}{1} x^{4-1} (1)^1 = 4 \cdot x^3 \cdot 1 = 4x^3
\]

3. For \(k = 2\):
\[
\binom{4}{2} x^{4-2} (1)^2 = 6 \cdot x^2 \cdot 1 = 6x^2
\]

4. For \(k = 3\):
\[
\binom{4}{3} x^{4-3} (1)^3 = 4 \cdot x^1 \cdot 1 = 4x
\]

5. 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:

\[
\boxed{x^4 + 4x^3 + 6x^2 + 4x + 1}
\]