Math Is Fun Forum

jacks

Guest

real roots

Prove that the equation

has no real roots.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: real roots

Hi;

The first thing you can do is use a Cauchy bound to determine that all real roots would have to be in the closed interval [-4,4].

Another improved bound proves the real roots if they exist are between [-2,2].

Interesting but non essential for the following method.

Form a Sturm chain:

Substitute two endpoints of of -∞ and ∞ into x.

You will get the 7 x 2 matrix

Count the number of sign changes in the first column and subtract the number of sign changes in the second column.

3 - 3 = 0

There are no real roots!

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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.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,646

Re: real roots

hi jacks,

I thought I'd start by looking at the graph y = F(x)

It has a single minimum as you can see and y > 0 for all x.

That's enough to show it has no real roots.

But a graph alone doesn't constitute a proof as it relies on a tool and doesn't show all values of x.

So one approach would be to prove that the graph truely has those properties.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

zetafunc.

Guest

Re: real roots

Hi;

The first thing you can do is use a Cauchy bound to determine that all real roots would have to be in the closed interval [-4,4].

How did you get that? With Cauchy's bound I am getting it in [-7,7]... since

where c is a root.

zetafunc.

Guest

Re: real roots

Sorry, (-7,7), not closed interval.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: real roots

Hi;

There are many variants of the Cauchy bound. I am using 2 of them:

http://en.wikipedia.org/wiki/Sturm%27s_theorem

and

http://www.mathsisfun.com/algebra/polyn … zeros.html

---

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.

jacks

Member

Registered: 2012-11-21

Posts: 132

Re: real roots

Thanks bobym and bob bundy

I have solved it like this way....

Now

and

So

So the equation has no real Roots.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: real roots

Hi jacks;

I forgot that idea. That is best.

---

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.

noelevans

Member

Registered: 2012-07-20

Posts: 236

Re: real roots

Hi!

Or if we evaluate f(-1) by the remainder theorem the signs in the quotient followed by the
remainder strictly alternate in sign:  + - + - + - + so there are no negative roots less than or
equal to -1.  (Lower bound theorem)

Similarly if we evaluate f(1) by the remainder theorem the signs in the quotient followed by
the remainder are all positive.  So there are no positive roots greater than or equal to 1.
(Upper bound theorem)

If |x|<=1 we have  x+3>=2.  Also the even powered terms cannot be negative.   Hence the
complete sum is >= 2 for all x in [-1,1]. 

Hence f(x)=0 has no real roots, and in fact is always positive since if there were a value a outside
of [-1,1] for which f(a)<0 then the intermediate value theorem would guarantee a root between
either 1 and a if a>1 or between -1 and a if a<-1.

Or more elegantly along the lines that Jack pointed out:  Completing the square on x^2+x+3 we get

x^2+x+3 = (x+1/2)^2 - 1/4 + 3 = (x+1/2)^2 + 11/4 which has minimum 11/4.

Also x^6 + x^4 is always >= zero.  Hence f(x) >= 11/4 (actually strictly >) for all x in the reals.
f(-1/2) = 181/65 = 2.828125.

Have a very blessed day!

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Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
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