Question

write out the binomial expansion of (a+b)^4 using the binomial theorem

Answer 1

The binomial theorem states that \((a + b)^n\) can be expanded as:

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

For \(n = 4\), the expansion of \((a + b)^4\) is given by:

\[
(a + b)^4 = \sum_{k=0}^{4} \binom{4}{k} a^{4-k} b^k
\]

Calculating each term in the expansion:

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

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

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

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

5. For \(k = 4\):
\[
\binom{4}{4} a^{4-4} b^4 = 1 \cdot 1 \cdot b^4 = b^4
\]

Now, combining all the terms, we have:

\[
(a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4
\]

Therefore, the binomial expansion of \((a + b)^4\) is:

\[
\boxed{a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4}
\]