This article will cover the binomial theorem, its formula, and what it is used for. The binomial theorem will then also be applied in examples to solve binomial expansions.
What is the Binomial Theorem Formula?
The binomial theorem allows us to expand an expression of the form into a polynomial sum containing x and y terms. A general formula for a binomial expression is given by,
.
which can be simplified to the following.
The binomial formula is the following
,
where both n and k are integers.
In the above expression, denotes the sum of all the terms starting at k = 0 until k = n.
Note that x and y can be interchanged here so the binomial theorem can also be written a
and this would give the same results as above.
The notation can be referred to as "n choose k" and gives a number called binomial coefficient, which is the number of different combinations of ordering k objects out of a total of n objects.
A binomial coefficient (n choose k, or ) is given by,
,
where "!" means factorial.
Factorial means the product of an integer with all the integers below it. For an integer n, we can express the factorial of n as
.
For example for 5 choose 3, we would have,
.
What is the proof for the binomial theorem?
The binomial theorem can be proved in a few different ways, but we will focus on the combinatorial proof.
Any expression in the form of can be written as
,
with a total of n products.
After expanding and removing the parenthesis, each term has the form of , for some arbitrary k between 0 and n.
The coefficient of this term has to be the number of ways to choose k values of y out of n values of . Therefore, the coefficient of is and
.
How to do a binomial expansion?
To understand how to expand a binomial expansion, we will look at an example.
Let's say we want to expand
. In this case, n = 4 and k will vary between 0 and 4. Using the formula for the binomial theorem, we can write:
.
We can now use the equation for the binomial coefficient to find all the constant terms in this expression. For the first term, 4 choose 0 (4C0), we have:
.
Repeating this for all 5 coefficients, we end up with binomial coefficients of 4C0 = 1, 4C1 = 4, 4C2 = 6, 4C3 = 4 and 4C4 = 1 in order.
Therefore, our expression for the binomial expansion simplifies to:
.
Note that y could also be replaced by any number.
Binomial theorem examples
The binomial theorem can also be used to find a specific term for a binomial expansion. For this, you do not have to carry out the whole expansion but will only be required to find one term. Let's look at an example to see how this can be done.
Find the coefficient of in the expansion of .
Solution
We recall the binomial theorem,
.
We notice that in our case n = 6, x = 2x and y = 1.
We need to find the term where the power of x is 4.
This will be when . So the term we are looking at in the formula is the third term
.
Replacing n = 6, y = 1 and x = 2x, we get
.
To find the binomial coefficient, we use
.
So the term we are looking for is
.As the question asks for the coefficient of the
term, the answer is simply 240.
Expand .
Solution
In this case, we have .
Using the binomial formula, we can expand this to
2)530+51(x2)431+52(x2)332+53(x2)233+54(x2)134+55(x2)035.
Now we need to calculate all the coefficients using the n choose k formula.
For we obtain
.
Repeating this for all coefficients, we get .
Therefore,
.
Binomial Theorem - Key takeaways