The Binomial theorem, also known as binomial expansion, explains the expansion of powers. It only applies to binomials. So, let’s start…
If we know that
And so on.
Notice that all addends come in the form . Numbers are called binomial coefficients. They are easily calculated and noted using factorials. Factorial function is a function that multiplies first n natural numbers. For natural number n with n! we denote the multiplication of first n natural numbers.
also, by the agreement.
Example 1. Calculate
First, when dealing with factorials never jump ahead and calculate everything because you can always somehow use the recursion. Now since we know that we have a common factor of al terms.
Example 2. Solve the following equation
Note: Second solution of this quadratic equation is removed because n has to be natural number.
If n is a natural number and k is a natural number or 0 and binomial coefficient is denoted with symbol and defined as:
The binomial coefficients have some specific properties.
Proof. When replacing k with 0 in the definition of binomial coefficient we get:
2. The symmetry property:
3. + = =
Example 3. Calculate using these properties.
= = =
Example 4. Calculate + using these properties.
+ = = =