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# Binomial Theorem

From weather forecast services to engineering and statistics, polynomial expressions are used daily to perform simple or more complicated mathematical operations. But how do you calculate a polynomial expression when the indices are very high and the calculations are long and tedious? This is where the binomial theorem comes in handy.

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 ${\left(x+y\right)}^{n}$ into a polynomial sum containing x and y terms. A general formula for a binomial expression is given by,

${\left(x+y\right)}^{n}=\left(\begin{array}{c}n\\ 0\end{array}\right){x}^{n}{y}^{0}+\left(\begin{array}{c}n\\ 1\end{array}\right){x}^{n-1}{y}^{1}+\left(\begin{array}{c}n\\ 2\end{array}\right){x}^{n-2}{y}^{2}+...+\left(\begin{array}{c}n\\ n-1\end{array}\right){x}^{1}{y}^{n-1}+\left(\begin{array}{c}n\\ n\end{array}\right){x}^{0}{y}^{n}$.

which can be simplified to the following.

The binomial formula is the following

${\left(x+y\right)}^{n}=\sum _{k=0}^{n}\left(\begin{array}{c}n\\ k\end{array}\right){x}^{n-k}{y}^{k}$,

where both n and k are integers.

In the above expression, $\sum _{k=0}^{n}$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

${\left(x+y\right)}^{n}=\sum _{k=0}^{n}\left(\begin{array}{c}n\\ k\end{array}\right){x}^{k}{y}^{n-k}$

and this would give the same results as above.

The notation $\left(\begin{array}{c}n\\ k\end{array}\right)$ 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 ${}^{n}C_{k}$) 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

$n!=n×\left(n-1\right)×\left(n-2\right)×...×1$.

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 ${\left(x+y\right)}^{n}$ can be written as

${\left(x+y\right)}^{n}=\left(x+y\right)×\left(x+y\right)×...×\left(x+y\right)$,

with a total of n products.

After expanding and removing the parenthesis, each term has the form of ${x}^{n-k}{y}^{k}$, 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 $\left(x+y\right)$. Therefore, the coefficient of ${x}^{n-k}{y}^{k}$ is $\left(\begin{array}{c}n\\ k\end{array}\right)$ and

${\left(x+y\right)}^{n}=\sum _{k=0}^{n}\left(\begin{array}{c}n\\ k\end{array}\right){x}^{n-k}{y}^{k}$.

## 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 ${x}^{4}$ in the expansion of ${\left(2x+1\right)}^{6}$.

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 ${x}^{n-2}={x}^{6-2}={x}^{4}$. So the term we are looking at in the formula is the third term

$\left(\begin{array}{c}n\\ 2\end{array}\right){x}^{n-2}{y}^{2}$.

Replacing n = 6, y = 1 and x = 2x, we get

$\left(\begin{array}{c}6\\ 2\end{array}\right){\left(2x\right)}^{4}{1}^{2}$.

To find the binomial coefficient, we use

$\left(\begin{array}{c}n\\ k\end{array}\right)=\frac{n!}{k!\left(n-k\right)!}=\frac{6!}{2!\left(4!\right)}=\frac{6×5×4×3×2×1}{2×1×4×3×2×1}=15$.

So the term we are looking for is $15×{\left(2x\right)}^{4}×{1}^{2}=240{x}^{4}$.As the question asks for the coefficient of the ${x}^{4}$ term, the answer is simply 240.

Expand ${\left({x}^{2}+3\right)}^{5}$.

Solution

In this case, we have $x={x}^{2},y=3andn=5$.

Using the binomial formula, we can expand this to

${\left({x}^{2}+3\right)}^{5}=\left(\begin{array}{c}5\\ 0\end{array}\right)\left({x}^{}$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 $\left(\begin{array}{c}5\\ 0\end{array}\right)$ we obtain

$\left(\begin{array}{c}5\\ 0\end{array}\right)=\frac{5!}{0!\left(5-0\right)!}=\frac{5×4×3×2×1}{1×5×4×3×2×1}=1$.

Repeating this for all coefficients, we get $\left(\begin{array}{c}5\\ 1\end{array}\right)=5,\left(\begin{array}{c}5\\ 2\end{array}\right)=10,\left(\begin{array}{c}5\\ 3\end{array}\right)=10,\left(\begin{array}{c}5\\ 4\end{array}\right)=5and\left(\begin{array}{c}5\\ 5\end{array}\right)=1$.

Therefore,

${\left({x}^{2}+3\right)}^{5}={x}^{10}+5{x}^{8}3+10{x}^{6}9+10{x}^{4}27+5{x}^{2}81+243={x}^{10}+15{x}^{8}+90{x}^{6}+270{x}^{4}+405{x}^{2}+243$.

## Binomial Theorem - Key takeaways

• The binomial theorem is used for carrying out binomial expansions. This means simplifying expressions of the form ${\left(x+y\right)}^{n}$into a polynomial sum in terms of x and y.
• The formula for the binomial theorem is:

• The simplified version of the binomial theorem is ${\left(x+y\right)}^{n}=\sum _{k=0}^{n}\left(\begin{array}{c}n\\ k\end{array}\right){x}^{n-k}{y}^{k}=\sum _{k=0}^{n}\left(\begin{array}{c}n\\ k\end{array}\right){x}^{k}{y}^{n-k}$
• The binomial coefficients or constant terms in this expression are found using:

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What is the binomial theorem used for?

The binomial theorem is used for carrying out binomial expansions. This means simplifying expressions of the form (x + y)n into a polynomial sum in terms of x and y.

What is the binomial theorem formula?

The simplified formula for the binomial theorem is (x + y)n = Σ (n choose k) xn–k yk.

How to expand an expression using the binomial theorem?

You can expand an expression by plugging the numbers into the binomial formula

(x + y)n = Σ (n choose k) xn–k yk

and working out the binomial coefficients with the formula (n choose k) = n! / [k!(n-k)!]

How to prove the binomial theorem?

The binomial theorem can be proved in a few different ways, but the most accessible may be the proof using combinatorics.

What is the binomial theorem?

The binomial theorem is the theorem that states the binomial formula.

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