We know that
\(
\begin{aligned}
& (a+b)^0=1 \\
& (a+b)^1=a+b \\
& (a+b)^2=a^2+2 a b+b^2
\end{aligned}
\)
and we can easily expand
\(
(a+b)^3=a^3+3 a^2 b+3 a b^2+b^3 .
\)
For higher powers, the expansion gets very tedious by hand! Fortunately, the Binomial Theorem gives us the expansion for and positive integer power of \((a+b)\) :
For any positive integer \({n}\),
\(
(a+b)^n=\sum_{k=0}^n\left(\begin{array}{c}
n \\
k
\end{array}\right) a^{n-k} b^k
\)
Where \(\left(\begin{array}{c}n \\ k\end{array}\right)={ }^n C_k=\frac{(n)(n-1)(n-2) \cdots(n-(k-1))}{k !}=\frac{n !}{k !(n-k) !}\)

Extensions of the Binomial Theorem

A useful special case of the Binomial Theorem is
\(
(1+x)^n=\sum_{k=0}^n\left(\begin{array}{l}
n \\
k
\end{array}\right) x^k
\)
for any positive integer \(n\), which is just the Taylor series for \((1+x)^n\).
This formula can be extended to all real powers \(\alpha\) :
\(
(1+x)^\alpha=\sum_{k=0}^{\infty}\left(\begin{array}{l}
\alpha \\
k
\end{array}\right) x^k
\)
for any real number \(\boldsymbol{\alpha}\), where
\(
\left(\begin{array}{l}
\alpha \\
k
\end{array}\right)=\frac{(\alpha)(\alpha-1)(\alpha-2) \cdots(\alpha-(k-1))}{k !}=\frac{\alpha !}{k !(\alpha-k) !}
\)
Notice that the formula now gives an infinite series: when \(\alpha=n\) is a positive integer, all but the first \((n+1)\) terms are 0 since after this \(
(n-n) ! = 0 !\) appears in each numerator.
This expansion is very useful for approximating \((1+x)^\alpha\) for \(|x| \ll 1\) :
\(
(1+x)^\alpha=1+\alpha x+\frac{\alpha(\alpha-1)}{2 !} x^2+\frac{\alpha(\alpha-1)(\alpha-2)}{3 !} x^3+\cdots
\)
But for \(|x| \ll 1\), higher powers of \(x\) get small very quickly, so \((1+x)^\alpha\) can be approximated to any accuracy we need by truncating the series after a finite number of terms.

Example 1:

For \(|x| \ll 1\),
\(
\begin{aligned}
(1+x)^{5 / 2} & \approx 1+\frac{5}{2} x \\
(1-2 x)^{100} & \approx 1-200 x \\
\left(1+x^2\right)^{-3} & \approx 1-3 x^2
\end{aligned}
\)
This type of reasoning is useful in investigating what happens when a physical system is perturbed slightly, introducing a new very small term \(\boldsymbol{x}\).

Example 2: By the binomial theorem,
\(
\begin{aligned}
(x+y)^3 & =\sum_{k=0}^3\left(\begin{array}{l}
3 \\
k
\end{array}\right) x^{3-k} y^k \\
& =\left(\begin{array}{l}
3 \\
0
\end{array}\right) x^3+\left(\begin{array}{l}
3 \\
1
\end{array}\right) x^2 y+\left(\begin{array}{l}
3 \\
2
\end{array}\right) x y^2+\left(\begin{array}{l}
3 \\
3
\end{array}\right) y^3 \\
& =x^3+3 x^2 y+3 x y^2+y^3
\end{aligned}
\)
as expected.