The Binomial Theorem states that:
\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
For \( n = 4 \), the expansion of \( (a + b)^4 \) is given by:
\[ (a + b)^4 = \sum_{k=0}^{4} \binom{4}{k} a^{4-k} b^k \]
Now, we calculate the coefficients and write out each term:
-
For \( k = 0 \): \[ \binom{4}{0} a^{4-0} b^0 = 1 \cdot a^{4} \cdot 1 = a^4 \]
-
For \( k = 1 \): \[ \binom{4}{1} a^{4-1} b^1 = 4 \cdot a^{3} \cdot b = 4a^3b \]
-
For \( k = 2 \): \[ \binom{4}{2} a^{4-2} b^2 = 6 \cdot a^{2} \cdot b^2 = 6a^2b^2 \]
-
For \( k = 3 \): \[ \binom{4}{3} a^{4-3} b^3 = 4 \cdot a^{1} \cdot b^3 = 4ab^3 \]
-
For \( k = 4 \): \[ \binom{4}{4} a^{4-4} b^4 = 1 \cdot 1 \cdot b^4 = b^4 \]
Putting it all together, the binomial expansion of \( (a + b)^4 \) is:
\[ (a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4 \]
1 answer