Polynomial Expansions (useful formulas)

Let $\mathrm{A}$ be the set of all " $\color{#3D99F6}{\text{useful}}$ " things.

Let $\mathrm{B}$ be the set of all " $\color{#20A900}{\text{Awesome}}$ " things.

Let $\mathrm{C}$ be the set of all " $\color{#D61F06}{\text{fascinating}}$ " things.

Let $\mathrm{D}$ be the set of all " $\color{#624F41}{\text{easily understandable}}$ " things.

What this note contains is an element of $\mathrm{A \cap B \cap C \cap D}$ ....

$\color{#3D99F6}{\textbf{Polynomial Expansions}}$

$\mathbf{1.}\quad \displaystyle \dfrac{1-x^{m+1}}{1-x} = 1+x+x^2+...+x^m = \sum_{k=0}^m x^k$

$\mathbf{2.} \quad \displaystyle \dfrac{1}{1-x} = 1+x+x^2+... = \sum_{k=0}^\infty x^k$

$\mathbf{3.}\quad \displaystyle (1+x)^n = 1+\binom{n}{1} x + \binom{n}{2}x^2+...+\binom{n}{n}x^n = \sum_{k=0}^n \dbinom{n}{k} x^k$

$\mathbf{4.}\quad \displaystyle (1-x^m)^n = 1-\binom{n}{1}x^m+\binom{n}{2}x^{2m}-...+(-1)^n\binom{n}{n} x^{nm}$

$\quad \quad \quad \quad\quad \displaystyle = \sum_{k=0}^n (-1)^n \dbinom{n}{k}x^{km}$

$\mathbf{5.}\quad \displaystyle\dfrac{1}{(1-x)^n} = 1+ \binom{1+n-1}{1} x + \binom{2+n-1}{2} x^2+...+\binom{r+n-1}{r} x^r+......$

$\quad \quad \quad \quad \quad \displaystyle =\sum_{k=0}^\infty \dbinom{k+n-1}{k} x^k$

$\color{#69047E}{\text{Tremendously useful}}$ in calculating the $\color{#3D99F6}{\text{co-efficient}}$ of any term in specially generating functions that we come across, in many combinatorics problems...

Taken From - Alan Tucker's "Applied Combinatorics"

Good luck problem solving !

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2 \times 3

2 \times 3

2^{34}

2^{34}

a_{i-1}

a_{i-1}

\frac{2}{3}

\frac{2}{3}

\sqrt{2}

\sqrt{2}

\sum_{i=1}^3

\sum_{i=1}^3

\sin \theta

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\boxed{123}

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(#Sharky_Surprises )

Aditya Raut - 5 years, 4 months ago

Easy Math Editor

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Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

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I know that the second one only works for $x<1$ . But do any of the others work for only $x>1$ . Also, thank you so much for this note, it's very useful.

Trevor Arashiro - 5 years, 5 months ago

-1<x<1 actually.

Bart Nikkelen - 5 years, 5 months ago

Thank you, but do u know which ones only work for this case

Can you add parts of this page into the algebra wiki? I think that Algebraic Identities and Algebraic Manipulation - Identities, would be suitable places to add them.

Calvin Lin Staff - 5 years, 3 months ago

what is the derivation of 5th one.

prashant goyal - 5 years, 5 months ago

Standard result, it's related to "Newton's generalised Binomial theorem", but if you do want the derivation please see it here

Aditya Raut - 5 years, 5 months ago

Definitely 2.

Eric Hernandez - 5 years, 5 months ago

Does the last one work for 0<x<1???

Tasneem Khaled - 5 years, 5 months ago

set contains Everything

Gautam Sharma - 5 years, 5 months ago

Is this for nerds like you?

Jack Daniel Zuñiga Cariño - 5 years, 5 months ago

Dude don't use nerds in a derogatory manner please. If you don't like nerds or aren't one yourself, you should remove yourself from Brilliant.org. Have a nice day.

Finn Hulse - 5 years, 5 months ago

@Finn Hulse , thanks for helping here, really !

img

I truly like your comment, by these many likes :-

@Cody Johnson :D

@Aditya Raut – Haha, anytime dude. :D

Agreed. BTW, some nerds can be good at sports as well.

Sharky Kesa - 5 years, 4 months ago

@Sharky Kesa –

@Sharky Kesa – BTW sharky, participate in JOMO 8, we miss your submission.

@Sharky Kesa , JOMO 8 starts $\color{#D61F06}{\textbf{TOMORROW}}$ and has some good questions I made.

This is for using in generating functions we design for combinatorics problems.... For example, see the set "vegetable combinatorics".... (type in search bar simply).... That's for all who want to learn, nothing high-figh technique or anything, just formulas to get co-efficient of a specific term in a generating function. @Jack Daniel Zuñiga Cariño

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$