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# Binomial Expansion

A binomial expansion is a method used to allow us to expand and simplify algebraic expressions in the form $$(x+y)^n$$ into a sum of terms of the form $$ax^by^c$$. If $$n$$ is an integer, $$b$$ and $$c$$ also will be integers, and $$b + c = n$$.

We can expand expressions in the form $$(x+y)^n$$ by multiplying out every single bracket, but this might be very long and tedious for high values of $$n$$ such as in $$(x+y)^{20}$$ for example. This is where using the Binomial Theorem comes in useful.

## The binomial theorem

The binomial theorem allows us to expand an expression of the form $$(x+y)^n$$ into a sum. A general formula for a binomial expression is:

$(x+y)^n = \binom{n}{0}x^ny^0 + \binom{n}{1}x^{n-1}y^1 + \binom{n}{2}x^{n-2}y^2 + \dots + \binom{n}{n-1}x^1y^{n-1} + \binom{n}{n}x^0y^n.$

Which can be simplified to:

\begin{align} (x+y)^n &= \sum\limits_{k=0}^n \binom{n}{k} x^{n-k}y^k \\ &= \sum\limits_{k=0}^n \binom{n}{k} x^ky^{n-k} . \end{align}

Where both $$n$$ and $$k$$ are integers. This is also known as the binomial formula. The notation

$\binom{n}{k}$

can be referred to as '$$n$$ choose $$k$$' and gives a number called the binomial coefficient which is the number of different combinations of ordering $$k$$ objects out of a total of $$n$$ objects. The equation for the binomial coefficient ($$n$$ choose $$k$$ or $$^nC_r$$ on a calculator) is given by:

$\binom{n}{k} = \frac{n!}{k!(n-k)!}$

Where '!' means factorial. Factorial means the product of an integer with all the integers below it. For example for $$5$$ choose $$3$$, we would have:

\begin{align} \binom{5}{3} &= \frac{5!}{3!(5-3)!} \\ &= \frac{5\cdot 4\cdot 3 \cdot 2 \cdot 1}{(3\cdot 2\cdot 1)(2\cdot 1)} \\ &= 10. \end{align}

## How do you do a binomial expansion?

To understand how to do a binomial expansion, we will look at an example. Let's say we want to expand $$(x+y)^4$$. In this case, $$n = 4$$ and $$k$$ will vary between $$0$$ and $$4$$. Using the formula for the binomial expansion, we can write:

$(x+y)^4 = \binom{4}{0}x^4y^0 + \binom{4}{1}x^3y^1 + \binom{4}{2}x^2y^2 + \binom{4}{3}x^1y^3+\binom{4}{4}x^0y^4.$

We can now use the equation for the binomial coefficient to find all the constant terms in this expression. For the first term we have:

\begin{align} \binom{4}{0} &= \frac{4!}{0!(4-0)!} \\ &= \frac{4 \cdot 3\cdot 2\cdot 1}{1\cdot (4 \cdot 3\cdot 2\cdot 1 )} \\ &= 1. \end{align}

Repeating this for all five coefficients, we end up with binomial coefficients of $$1$$, $$4$$, $$6$$, $$4$$, $$1$$ in order. Therefore, our expression for the binomial expansion simplifies to:

$x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4.$

Note that $$y$$ could also be replaced by any number.

## Binomial expansion for fractional and negative powers

Sometimes you will encounter algebraic expressions where n is not a positive integer but a negative integer or a fraction. Let's consider the expression $$\sqrt{1-2x}$$ which can also be written as

$(1- 2x)^\dfrac{1}{2}$ where $$x < 0.5$$. In this case, it becomes hard to find the formula to find the binomial coefficients,

because we can't find the factorials for a negative or rational number. However, if we look at an example for a positive integer, we can find a more general expression that we can then also apply to negative and fractional numbers. For example for

$\binom{6}{3}$

we have

\begin{align} \binom{6}{3}&= \frac{6!}{3!(6-3)!} \\ &= \frac{6\cdot 5\cdot 4}{3!} \\ &= \frac{6(6-1)(6-2)}{3!}. \end{align}

From this we observe that

$\binom{n}{k} = \frac{n(n-1)(n-2)(n-3)\dots (n-k+1)}{k!}$

and therefore the more general expression for the binomial theorem is the infinite formula

$(a+b)^n = \frac{a^n}{0!} + \frac{na^{n-1}b}{1!} + \frac{n(n-1)a^{n-2}b^2}{2!} + \frac{n(n-1)(n-2)a^{n-3}b^3}{3!} + \dots$

Let's look at $$\sqrt{1-2x}$$. In this case $$a = -2x$$, $$b = 1$$ and $$n =1/2$$. Substituting this we get:

\begin{align} \frac{(-2x)^\frac{1}{2}}{0!} &+ \frac{\left(-\frac{1}{2}\right) (-2x)^{-\frac{1}{2}}\cdot 1 }{1!} \\ &\quad + \frac{\left(-\frac{1}{2}\right) \left(-\frac{1}{2}\right) (-2x)^{-\frac{3}{2}}\cdot 1^2 }{2!} \\ &\quad + \frac{\left(-\frac{1}{2}\right) \left(-\frac{1}{2}\right) \left(-\frac{3}{2}\right) (-2x)^{-\frac{5}{2}}\cdot 1^3 }{3!} + \dots \end{align}

Using Mac Laurin's expansion we can say that the above expression converges to

$\sqrt{1-2x} = 1 - x - \frac{x^2}{2} - \frac{x^3}{2}.$

## Binomial Expansion - Key takeaways

• A binomial expansion helps us to simplify algebraic expressions into a sum
• The formula for the binomial expansion is:

$(x+y)^n = \binom{n}{0}x^ny^0 + \binom{n}{1}x^{n-1}y^1 + \binom{n}{2}x^{n-2}y^2 + \dots + \binom{n}{n-1}x^1y^{n-1} + \binom{n}{n}x^0y^n.$

• The binomial coefficients or constant terms in this expression are found using:$\binom{n}{k} = \frac{n!}{k!(n-k)!}$
• $(1+a)^n = 1 + na+ \frac{n(n-1)}{2!}a^2 + \frac{n(n-1)(n-2)}{3!}a^3 + \dots$

The constant term is found by using the formula

n choose k=n!/k!(n-k)!

A binomial expansion is a method that allows us to simplify complex algebraic expressions into a sum.

You can use the binomial expansion formula

(x+y)^n=(nC0)x^n y^0+(nC1)x^/n-1)y^1+(nC2)x^(n-2)y^2+...+(nCn-1)x^1y^(n-1)+(nCn)x^0y^n

## Final Binomial Expansion Quiz

Question

What is the binomial expansion used for?

Expanding out things like $$(x+y)^n$$ without having to do all the multiplication.

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Question

What is the simplified form of the binomial expansion formula using summation notation?

$(x+y)^n = \sum\limits_{k=0}^n {n\choose k} x^{n-k}y^k.$

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Question

How do you find the binomial coefficients?

Using $$n\choose k$$.

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Question

What is 6 choose 3?

20.

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Question

What is 7 choose 4?

35.

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Question

What is 5 choose 0?

1.

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Question

What is 6 choose 6?

1.

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Question

Expand $$(1+2x)^5$$.

$$1+10x+40x^2+80x^3+80x^4+32x^5$$.

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Question

Which of these is the correct expansion of $$(x^2-y)^2$$?

$$x^4-2x^2y+y^2$$.

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Question

Find the first three terms in the expansion of $$(4+5x)^{\frac{1}{2}}$$.

$$2+\dfrac{5}{4}x -\dfrac{25}{64}x^2$$.

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Question

Find the first three terms in the expansion of $$(1-2x)^{-\frac{1}{2}}$$.

$$1+x+\dfrac{3}{2}x^2$$.

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Question

Given that the coefficient of $$x^2$$ in the expansion of $$(1+ax)^7$$ is $$525$$, find the possible values of $$a$$.

$$5$$.

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Question

Which of these is equivalent to $$n!$$?

$$n (n-1)\dots (2)(1)$$.

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Question

Which of these is the correct expansion of $$6 choose 2$$?

$$\dfrac{6!}{2!4!}$$.

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Question

How do you read the notation $$n \choose k$$?

$$n$$ choose $$k$$.

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Question

What does the notation $$n\choose k$$ mean?

$$\dfrac{n!}{(n-k)!k!}$$.

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Question

Functions in mathematics with the symbol (!) that multiply a number by every number that precedes it are…?

Factorials

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Question

What is the factorial notation?

n!. Where n is a positive integer.

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Question

The rule says that the factorial of any number is that number times the factorial of (that number minus 1). n! = n × (n−1) is called…?

Factorial rule

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Question

What is 0!?

0! = 1

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Question

What are factorials used for?

Arrangements, permutations, and combinations.

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Question

What is the factorial of 9?

9! =9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 362,880

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Question

How many possible combinations can you make with a four-digit number?

4 × 3 × 2 × 1 = 26

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Question

How do you calculate factorials?

Multiply your number by every number below it.

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Question

Evaluate 3!7!

3!7! = (3× 2× 1) (7 × 6 × 5 × 4 × 3 × 2 × 1) = 30240

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Question

What can Pascal´s triangle help us find?

Binomial coefficients.

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Question

What is the $$n$$ number for the first row in Pascal´s triangle?

$$n=0$$.

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Question

What is the first value in Pascal's triangle?

$$1$$.

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Question

What are the values on the extremes of each row?

$$1$$.

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Question

How do you find the values of a row?

By adding together the values above it.

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Question

How many elements are in row $$n$$ of Pascal's triangle?

$$n+1$$.

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Question

What is the formula for finding the sum of each row?

$$2^n$$.

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Question

How do you find Fibonacci´s sequence in Pascal's triangle?

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Question

What are the first five values of Fibonacci's sequence?

1, 1, 2, 3, 5.

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Question

What is the sum of the elements in the 8th row ($$n=7$$) of Pascal's triangle?

$$128$$.

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Question

What are the binomial coefficients for the expansion $$(x+y)^6$$?

$$1, 6, 15, 20, 15, 6, 1$$.

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Question

What are the binomial coefficients for the expansion $$(x+2y)^4$$.

$$1, 4, 6, 4, 1$$.

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Question

How do you use Pascal's Triangle to find the Fibonacci numbers?

Sum along the diagonals.

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Question

If you are making Pascal's triangle from scratch, what three numbers do you put in a triangle first?

$$1$$ in all three spots.

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Question

If you want to find the next row of Pascal's triangle, what do you do to the elements of the previous row?

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