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 March 7th, 2017, 01:31 AM #1 Newbie   Joined: Sep 2016 From: Maharashtra Posts: 8 Thanks: 0 inverting polynomial Hi My efforts to find an inverse for this equation: f(x) = (3x^3 + 6x^2 + 7x), exists or not are in vain. Could anyone help me try to solve this. I would be grateful.
 March 7th, 2017, 11:29 AM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 1,944 Thanks: 1011 $f(x) = 3x^3 + 6x^2 + 7x$ $f'(x) = 9x^2 + 12x + 7$ Functions with inverses must be monotonic over their natural or restricted domain. Is $f(x)$ monotonic on $\mathbb{R}$? How might you tell? Thanks from topsquark and classkid
 March 7th, 2017, 05:17 PM #3 Newbie   Joined: Sep 2016 From: Maharashtra Posts: 8 Thanks: 0 Actually the f(x) is over Zn where n = 9. So the problem is to find if f(x) mod 9 has an inverse or not.
March 7th, 2017, 05:24 PM   #4
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Quote:
 Originally Posted by classkid Actually the f(x) is over Zn where n = 9. So the problem is to find if f(x) mod 9 has an inverse or not.
That little piece of info should have been in the first post.

You folks just love to waste our time.

 March 8th, 2017, 01:49 AM #5 Newbie   Joined: Sep 2016 From: Maharashtra Posts: 8 Thanks: 0 I am sorry. But could you please help me solving this problem?
 March 8th, 2017, 08:06 AM #6 Member   Joined: Jan 2016 From: Athens, OH Posts: 88 Thanks: 47 The following table of values of the 9 elements mod 9 shows that the function is indeed both surjective and injective. Hence it does have an inverse. Thanks from topsquark and classkid
 March 8th, 2017, 10:05 AM #7 Global Moderator   Joined: Dec 2006 Posts: 18,965 Thanks: 1606 3x² + x
 March 8th, 2017, 05:31 PM #8 Newbie   Joined: Sep 2016 From: Maharashtra Posts: 8 Thanks: 0 So if I want to find its inverse in an equation form, what would be my approach? Could u please tell me.
March 8th, 2017, 10:34 PM   #9
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Quote:
 Originally Posted by skipjack 3x² + x
I'd be interested in knowing how you came up with this.

It's not unique. There are two other "inverses".

$3x^3 + 3x^2 + 7x$

$6x^3 + 3x^2 + 4x$

I determined these through the most brutish of force.

 March 8th, 2017, 11:37 PM #10 Global Moderator   Joined: Dec 2006 Posts: 18,965 Thanks: 1606 I just looked for a quadratic and found one. Once a solution is known, others follow as 3x(x - 1)(x + 1) ≡ 0 (mod 9). Thanks from classkid

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