Math Is Fun Forum

yahoopi

Guest

Polynomial Challenge ONE

Need help - who can do this? This is just crazy. I can deal with cubes but this is just... ugh how??

x¹⁵ − 3x¹⁴ − 2x¹³ − 6x¹² − 34x¹¹ + 18x¹⁰ − 40x⁹ − 48x⁸ + 5x⁷ − 87x⁶ + 26x⁵ + 54x⁴ + 28x³ + 72x² + 16x = 0

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Polynomial Challenge ONE

Well, here's what that function looks like.

From the graph, you can make some guesses at some of the solutions.
Once you've checked that a solution x=k is correct, you can then take a factor of (x-k) out of the function and continue analysing the rest of it.

Hopefully, once you've found all the "easy" roots, the remaining part of the function will be easier to look at in more detail.

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Why did the vector cross the road?
It wanted to be normal.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Polynomial Challenge ONE

Hi yahoopi;

Since apparently I like this equation better, I solved it for you. If you ever need it...
The one you are interested in is below.

You can't do big ones like that by hand. The story goes it took Alan Turing a whole month to find the roots of 7th order poly to 6 digits. Imagine how long this one would take.

After you deflate out the real roots, use newtons iteration to the complex plane for the rest. Or use a CAS.

x= -1
x= 0
x= 1
x= -1.879479256367186
x= -1.026622372345294
x= -0.2343788141426711
x= 3.929910628259091

x= -0.06692366674117192 - 1.03460975394155 i
x= -0.06692366674117192 + 1.03460975394155 i
x= 0.05147872539460732 - 0.9566009836684153 i
x= 0.05147872539460732 + 0.9566009836684153 i
x= 0.1391601029722258 - 2.117631580642914 i
x= 0.1391601029722258 + 2.117631580642914 i
x= 0.9815697456723689 - 1.030956164219834 i
x= 0.9815697456723689 + 1.030956164219834 i

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Polynomial Challenge ONE

Bobby, could you check that you copied the function correctly?
I work out x=-2 to be a solution, so one of us made an error somewhere.

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Why did the vector cross the road?
It wanted to be normal.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Polynomial Challenge ONE

Hi mathsyperson;

Curse all word processors! Especially Open Office! Of course you are right, darn copy and paste failed to copy correctly just 1 of the exponents.

x¹⁵ − 3x¹⁴ − 2x¹³ − 6x¹² − 34x¹¹ + 18x¹⁰ − 40x⁹ − 48x⁸ + 5x⁷ − 87x⁶ + 26x⁵ + 54x⁴ + 28x³ + 72x² + 16x = 0

This is what I solved for now. Hope they are the same!!!!

These are the roots:

This one is obviously more manageable and you might be able to get them by hand.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.