Upper and Lower Bounds of Polynomials

[opus]

Say we have a polynomial $f(x)=2x^3+3x^2-14x-21$ and we want to find the upper and lower bounds of the real zeros of this polynomial.

If no real zero of $f$ is greater than b, then b is considered to be the upper bound of $f$. And if no real zero of $f$ is less than a, then a is considered to be the lower bound.

Now to find the upper and lower bounds of $f(x)=2x^3+3x^2-14x-21$, my book uses synthetic division of $f(x)=2x^3+3x^2-14x-21$ by the numbers 2 and 3 for the upper bound (2 didn't give an upper bound but 3 did). And synthetic division by the numbers -3 and -4 (-3 didn't give a lower bound but -4 did) for the lower bound. Where are these numbers come from? Do you just start at some low integer and keep plugging new ones into the divisor until you get the desired quotient (all positive coefficients for upper bound, and alternating signs for lower bound)?

The book mentions that by testing a small positive number, and progressively testing larger ones, this will find the smallest upper bound and the largest lower bound. How is there are smallest upper bound and largest lower bound? It would only make sense to have a single upper bound and a single lower bound.

 

[Gene Naden]

If no real zero of fff is greater than b, then b is considered to be the upper bound of f

It would only make sense to have a single upper bound and a single lower bound.

By the definition you gave, the upper bounds form a set starting with the smallest upper bound and going up from there.

Can you show what you divided by in your synthetic division?

 

[StoneTemplePython]

Say we have a polynomial $f(x)=2x^3+3x^2-14x-21$ and we want to find the upper and lower bounds of the real zeros of this polynomial.

If no real zero of $f$ is greater than b, then b is considered to be the upper bound of $f$. And if no real zero of $f$ is less than a, then a is considered to be the lower bound.

Now to find the upper and lower bounds of $f(x)=2x^3+3x^2-14x-21$, my book uses synthetic division of $f(x)=2x^3+3x^2-14x-21$ by the numbers 2 and 3 for the upper bound (2 didn't give an upper bound but 3 did). And synthetic division by the numbers -3 and -4 (-3 didn't give a lower bound but -4 did) for the lower bound. Where are these numbers come from? Do you just start at some low integer and keep plugging new ones into the divisor until you get the desired quotient (all positive coefficients for upper bound, and alternating signs for lower bound)?

Going back to your other thread, the first thing you should try to do is write

$f(x) =2 h(x)= 2\big( x^3 + \frac{3}{2}x^2 -7x - \frac{21}{2}\big)$

To be honest, I wouldn't dwell on this particular bounding process too much. The takeaways here are (a) how does synthetic division work, (b) what does all coefficients positive tell you, and (c) what about mixed signs in front of coefficients.

In general complex numbers are fundamental for roots of polynomials -- most of the bounds on roots that I'm aware of look at the complex plane and bound them to be in some disc (i.e. circle with radius $r$ centered at some point, and shading it in).

As for where they come from? If the author didn't tell you, I have a hunch its because the author knew the roots are

$x_1 = \sqrt{7}$
$x_2 = - \sqrt{7}$
$x_3 = -1.5$

and given that insider knowledge of the roots, worked backwards to find bounds like 3. note all roots are $\in [-3,3]$ or if you want to tighten it $[-\sqrt{7},\sqrt{7}]$

The book mentions that by testing a small positive number, and progressively testing larger ones, this will find the smallest upper bound and the largest lower bound. How is there are smallest upper bound and largest lower bound? It would only make sense to have a single upper bound and a single lower bound.

suppose I have a variable $x \in [0,1]$. You could say $x \leq 1$. This would be a tight upper bound. You could also say $x \leq 701214.12312$. This would be, well, a looser upper bound.

There's some linguistic problems remaining though. If synthetic division of $\big(x- (-3)\big)$ doesn't give a lower bound but doing $\big(x- (-4\big)$ does, then perhaps the text is referring to the tightest bound that can be achieved given this process. In general smallest upper bound is of course $\sqrt{7}$ and the largest lower bound is $-\sqrt{7}$, though I am clearly working backward from knowledge of the roots to say this.

 

[opus]

@Gene Naden
Attached screenshot of a the beginnings of the problem I'm referring to.

 

[opus]

Going back to your other thread, the first thing you should try to do is write

$f(x) =2 h(x)= 2\big( x^3 + \frac{3}{2}x^2 -7x - \frac{21}{2}\big)$

To be honest, I wouldn't dwell on this particular bounding process too much. The takeaways here are (a) how does synthetic division work, (b) what does all coefficients positive tell you, and (c) what about mixed signs in front of coefficients.

In general complex numbers are fundamental for roots of polynomials -- most of the bounds on roots that I'm aware of look at the complex plane and bound them to be in some disc (i.e. circle with radius $r$ centered at some point, and shading it in).

As for where they come from? If the author didn't tell you, I have a hunch its because the author knew the roots are

$x_1 = \sqrt{7}$
$x_2 = - \sqrt{7}$
$x_3 = -1.5$

and given that insider knowledge of the roots, worked backwards to find bounds like 3. note all roots are $\in [-3,3]$ or if you want to tighten it $[-\sqrt{7},\sqrt{7}]$
suppose I have a variable $x \in [0,1]$. You could say $x \leq 1$. This would be a tight upper bound. You could also say $x \leq 701214.12312$. This would be, well, a looser upper bound.

There's some linguistic problems remaining though. If synthetic division of $\big(x- (-3)\big)$ doesn't give a lower bound but doing $\big(x- (-4\big)$ does, then perhaps the text is referring to the tightest bound that can be achieved given this process. In general smallest upper bound is of course $\sqrt{7}$ and the largest lower bound is $-\sqrt{7}$, though I am clearly working backward from knowledge of the roots to say this.

Ok now I understand the tight vs loose bounds. It's just a matter of narrowing the set of numbers down.

From a standpoint outside of the text, without the foresight of the author to know what numbers to plug into the synthetic division, how do we know what to do synthetic division by to find the upper and lower bounds? Initially I thought the author was dividing by numbers that were found in the possible rational zeros test, but 4 was not among that set of numbers. Are we really just starting with a low integer, and plugging them in until we get all a quotient with all positive coefficients and a quotient with alternating signs?

 

[StoneTemplePython]

From a standpoint outside of the text, without the foresight of the author to know what numbers to plug into the synthetic division, how do we know what to do synthetic division by to find the upper and lower bounds? Initially I thought the author was dividing by numbers that were found in the possible rational zeros test, but 4 was not among that set of numbers. Are we really just starting with a low integer, and plugging them in until we get all a quotient with all positive coefficients and a quotient with alternating signs?

quite possibly. Is this an assigned text? If you author gives a procedure (algorithm) but only via an opaque example, it's not a great text. Put differently, to 'teach' someone an algorithm and not tell them how it is initialized... isn't really an algorithm.

What I said about focusing on (a), (b) and (c) stands.

 

[opus]

Ok, and do we include this bounds in the solution set? That is, say I synthetically divide a polynomial by 4, and this is found to be the least upper bound, and i do the same process to find -1 to be the greatest lower bound. The text lists these bounds as [-1,4], however neither of these divisors gave a remainder of 0, so they can't be in the solution set, yet the brackets imply that they are included.

 

[StoneTemplePython]

Ok, and do we include this bounds in the solution set? That is, say I synthetically divide a polynomial by 4, and this is found to be the least upper bound, and i do the same process to find -1 to be the greatest lower bound. The text lists these bounds as [-1,4], however neither of these divisors gave a remainder of 0, so they can't be in the solution set, yet the brackets imply that they are included.

it's the other way around.

your real roots to the polynomial are any $x_k \in \mathbb R$ such that $f(x_k) = 0$, right? Well this bounding process is narrowing the scope to say all real $x_k \in [-1,4]$ as opposed to being anywhere on the real line.

edit:
to be crystal clear, you have now changed examples and are talking about a different problem, correct? For your earlier example I explicitly stated $x_2 \lt -1$ and ditto for $x_3$ -- i.e. I stated $x_2, x_3 \not \in [-1,4]$ so this must be a bound for a different problem.

 

[opus]

Ok that makes sense, thank you.

And yes, it was a different problem. I just made up some easy numbers to ask a different question relating to the same thing.