The Binomial Theorem is a fundamental theorem in algebra that provides a formula to expand powers of binomials. It allows us to easily expand expressions,
, where
and
are real numbers or variables, and
.
Proposition: For any real numbers
and
Where
Proof (Mathematical Induction):
For
For
For
For
Let's assume
for some
Now we just have to show that the equation holds true for
Or,
∴