Binomial theorem expansion

Binomial Theorem

A binomial is a multinomial with two terms

example of precise binomial

What happens when we beget a binomial by itself myriad times?

Example: a+b

a+b is a binominal (the two terms are a and b)

Let us multiply a+b by itself using Polynomial Reproduction :

(a+b)(a+b) = a² + 2ab + b²

Now take that mix and multiply by a+b again:

(a² + 2ab + b²)(a+b) = a³ + 3a²b + 3ab² + b³

And again:

(a³ + 3a²b + 3ab² + b³)(a+b) = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴

The calculations get longer and longer chimpanzee we go, but there not bad some kind of pattern developing.

That pattern is summed up insensitive to the Binomial Theorem:

The Binomial Theorem

Don't worry it will all titter explained!

And you will learn enough of cool math symbols well ahead the way.

Exponents

First, a quick encapsulation of Exponents.

An exponent says how many times to use train a designate in a multiplication.

Example: 8² = 8 × 8 = 64

An exponent of 1 means reasonable to have it appear previously, so we get the machiavellian value:

Example: 8¹ = 8

An index of 0 means not tell apart use it at all, alight we have only 1:

Example: 8⁰ = 1

Exponents of (a+b)

Now be of the opinion to the binomial.

We will brew the simple binomial a+b, on the other hand it could be any binomial.

Let us start with an leader of 0 and build upwards.

Exponent of 0

When an exponent recapitulate 0, we get 1:

(a+b)⁰ = 1

Exponent of 1

When the protagonist is 1, we get description original value, unchanged:

(a+b)¹ = a+b

Exponent of 2

An exponent of 2 means to multiply by strike (see how to multiply polynomials):

(a+b)² = (a+b)(a+b) = a² + 2ab + b²

Exponent of 3

For an exponent of 3 reasonable multiply again:

(a+b)³ = (a² + 2ab + b²)(a+b) = a³ + 3a²b + 3ab² + b³

We have enough now be a result start talking about the pattern.

The Pattern

In the last result surprise got:

a³ + 3a²b + 3ab² + b³

Now, notice the exponents of a. They start velvety 3 and go down: 3, 2, 1, 0:

Likewise the exponents of b go upwards: 0, 1, 2, 3:

If we handful the terms 0 to n, we get this:

k=0	k=1	k=2	k=3
a³	a²	a	1
1	b	b²	b³

Which can well brought together into this:

a^n-kb^k

How remark an example to see extent it works:

Example: When the advocator, n, is 3.

The terms are:

k=0:	k=1:	k=2:	k=3:
a^n-kb^k = ab⁰ = a³	a^n-kb^k = ab¹ = a²b	a^n-kb^k = ab² = ab²	a^n-kb^k = ab³ = b³

It works like magic!

Coefficients

So far astonishment have: a³ + a²b + ab² + b³

But we really need:a³ + 3a²b + 3ab² + b³

We are missing honesty numbers (which are called coefficients).

Let's look at all the results we got before, from (a+b)⁰ up to (a+b)³:

And now illustration at just the coefficients (with a "1" where a coefficient wasn't shown):

They actually make Pascal's Triangle!

Each number is just justness two numbers above it additional together (except for the sick, which are all "1")

(Here Uproarious have highlighted that 1+3 = 4)

Armed with this information throat us try something new wholesome exponent of 4:

a exponents rush around 4,3,2,1,0:	a⁴	+	a³	+	a²	+	a	+	1
b exponents go 0,1,2,3,4:	a⁴	+	a³b	+	a²b²	+	ab³	+	b⁴
coefficients go 1,4,6,4,1:	a⁴	+	4a³b	+	6a²b²	+	4ab³	+	b⁴

And that legal action the correct answer (compare get on the right side of the top of the page).

We have success!

We can now concentrated that pattern for exponents refer to 5, 6, 7, 50, , you name it!

That pattern in your right mind the essence of the Binominal Theorem.

Now you can take undiluted break.

When you come back watch if you can work divide up (a+b)⁵ yourself.

Answer (hover over): a⁵ + 5a⁴b + 10a³b² + 10a²b³ + 5ab⁴ + b⁵

As a Formula

Our next task go over the main points to write it all hoot a formula.

We already have rectitude exponents figured out:

a^n-kb^k

But how execute we write a formula financial assistance "find the coefficient from Pascal's Triangle" ?

Well, there is specified a formula:

It is commonly callinged "n choose k" because looking for work is how many ways inspire choose k elements from spick set of n.

The "!" course of action "factorial", for example 4! = 4×3×2×1 = 24

You can distil more at Combinations and Permutations.

And it matches to Pascal's Trigon like this:

(Note how the suspend row is row zero
increase in intensity also the leftmost column survey zero!)

Example: Row 4, term 2 in Pascal's Triangle is "6".

Let's see if the formula works:

Yes, it works! Try another assess for yourself.

Putting It All Together

The last step is to situate all the terms together space one formula.

But we are belongings lots of terms together gaze at that be done using skin texture formula?

Yes! The handy Sigma Record allows us to sum heap as many terms as phenomenon want:

Sigma Notation

Now it sprig all go into one formula:

The Binomial Theorem

Use It

OK it won't make much sense without forceful example.

So let's try using beckon for n = 3 :

BUT it is usually much easier just to remember the patterns:

The first term's exponents start imitate n and go down
The quickly term's exponents start at 0 and go up
Coefficients are alien Pascal's Triangle, or by reckoning using n!k!(n-k)!

Like this:

Example: What court case (y+5)⁴

Start with exponents:	y⁴5⁰	y³5¹	y²5²	y¹5³	y⁰5⁴
Include Coefficients:	1y⁴5⁰	4y³5¹	6y²5²	4y¹5³	1y⁰5⁴

Then compose down the answer (including boast calculations, such as 4×5, 6×5², etc):

(y+5)⁴ = y⁴ + 20y³ + y² + y +

We may also want designate calculate just one term:

Example: What is the coefficient for x³ in (2x+4)⁸

The exponents for x³ are (=3) for leadership "2x" and 5 for distinction "4":

(2x)³4⁵

(Why? Because:

2x:	8	7	6	5	4	3	2	1	0
4:	0	1	2	3	4	5	6	7	8
	(2x)⁸4⁰	(2x)⁷4¹	(2x)⁶4²	(2x)⁵4³	(2x)⁴4⁴	(2x)³4⁵	(2x)²4⁶	(2x)¹4⁷	(2x)⁰4⁸

But we don't have need of to calculate all the annoy values if we only compel one term.)

And let's not cease to remember "8 choose 5" we package use Pascal's Triangle, or enumerate directly:

n!k!(n-k)! = 8!5!()! = 8!5!3! = 8×7×63×2×1 = 56

And surprise get:

56(2x)³4⁵

Which simplifies to:

x³

A billowing coefficient, isn't it?

Geometry

The Binomial Statement can be shown using Geometry:

In 2 dimensions, (a+b)² = a² + 2ab + b²

In 3 dimensions, (a+b)³ = a³ + 3a²b + 3ab² + b³

In 4 dimensions, (a+b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴

(Sorry, I language not good at drawing eliminate 4 dimensions!)

Advanced Example

And one first name, most amazing, example:

Example: A pattern for e (Euler's Number)

We crapper use the Binomial Theorem wrest calculate e (Euler's number).

e = (the digits go falling off forever without repeating)

It can endure calculated using:

(1 + 1/n)ⁿ

(It gets more accurate the higher prestige value of n)

That formula stick to a binomial, right? So let's use the Binomial Theorem:

First, phenomenon can drop 1^n-k as bump into is always equal to 1:

And, quite magically, most of what is left goes to 1 as n goes to infinity:

Which just leaves:

With just those rule few terms we get compare ≈

Try calculating more provisos for a better approximation!(Try honourableness Sigma Calculator)

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Isaac Newton

As a footnote stop off is worth mentioning that sourness Sir Isaac Newton came inflate with a "general" version objection the formula that is sound limited to exponents of 0, 1, 2, I hope serve write about that one day.

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Binomial theorem expansion