remember the preliminaries!

I am unable to prove 2 statements. Please help
1)(a+b)^n - a^n - b^n is always divisible by ab for all n belongs to N.
2) a polynomial of odd degree will always have one of its roots to be either +1 or -1.

#Polynomialproofs

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

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}`	$\boxed{123}$

Comments

For statement 1), use the binomial theorem to find that

$(a + b)^{n} = a^{n} + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^{2} + .... + \binom{n}{n-1}ab^{n-1} + b^{n}.$

After subtracting $a^{n}$ and $b^{n}$ the remaining terms are all divisible by $ab,$ and thus so is their sum.

Statement 2) is not actually true. $f(x) = x - 2$ is of odd degree and only has root $2.$

Brian Charlesworth - 6 years ago

sir , I have a problem . I very much want to study number theory but I am unable to grasp the concepts beyond modular multiplication . I earnestly want to learn those Fermat rules ,Euler theorem , CRT etc,etc, Please help.

Raven Herd - 6 years ago

Have you tried the wikis here on Brilliant.org? For example, here is the one on the CRT.

This can be found by choosing "Topics", followed by "Number Theory", "Modular Arithmetic" and then the 'open book' icon to the right of "Chinese Remainder Theorem". You can find the other topics in your list in this fashion as well.

Brian Charlesworth - 6 years ago

@Brian Charlesworth – yes I have tried them. I devoted full 1 week and I was quite confident that I can learn it . But it didn't work. Please reply soon. P.S. Sir, an odd request : Where is the started problems option? I am hunting it since the set up of Brilliant has changed.

Raven Herd - 6 years ago

@Raven Herd – I'm not sure what to suggest next. If you spent a week going through all the wikis then you probably know more than I do now about this topic. Sometimes it's just about practice, so keep trying Number Theory questions on Brilliant (and elsewhere) and maybe the concepts will eventually become second nature. Sorry I couldn't suggest something more helpful. :(

As for the "Started Problems" option, I noticed it missing from the main page a few days ago as well. I eventually found it, though. Click on the "blue planet" icon and choose the "View mobile site" option. Once on that page, click on the "three dot" icon in the upper right corner and you will see the "Started problems" option listed there. Then, if you want, you can choose the "View full site" option on the same list to take your list of started problems back to the main site format.

Brian Charlesworth - 6 years ago

@Brian Charlesworth – Sir ,you are too modest.I have decided that I will go through the wikis again.I believe I have not been sincere and Brilliant wikis are the best study material I have come across till now and the best part about them is that they are prepared by people who find a particular topic fascinating and thus eliminate the chances of errors greatly. Anyways, thanks for you consideration.

Raven Herd - 6 years ago

@Raven Herd – You're welcome. Good luck with your studies. :)

Brian Charlesworth - 6 years ago

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}`	$\boxed{123}$