The expression \((k+1)^4\) can be expanded using the binomial theorem. The binomial theorem states that:

\[
(a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r
\]

For \((k+1)^4\), we have \(a = k\), \(b = 1\), and \(n = 4\).

So, the expansion is:

\[
(k+1)^4 = \binom{4}{0}k^4 + \binom{4}{1}k^3(1) + \binom{4}{2}k^2(1^2) + \binom{4}{3}k(1^3) + \binom{4}{4}(1^4)
\]

Simplifying the binomial coefficients:

\[
= 1 \cdot k^4 + 4 \cdot k^3 + 6 \cdot k^2 + 4 \cdot k + 1
\]

Thus, the expanded form of \((k+1)^4\) is:

\[
k^4 + 4k^3 + 6k^2 + 4k + 1
\]

Would you like further details on binomial expansion or another concept?

### Related Questions:
1. What is the general form of the binomial theorem for any power?
2. How can binomial expansion be applied in calculus, such as in differentiation or integration?
3. How do we compute binomial coefficients using factorials?
4. What is the expansion of \((k+2)^4\)?
5. How would you expand \((k+1)^5\)?

**Tip:** The binomial theorem is useful for expanding polynomials without needing to multiply them repeatedly!

Explore the expansion of (k+1)^4 using the binomial theorem. Learn the key steps to apply binomial coefficients and simplify the terms to get the expanded form k^4 + 4k^3 + 6k^2 + 4k + 1. Ideal for high school students studying algebra and polynomial expansions.

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