SOLUTION: Use the upper and lower bound theorem find the smallest positive integer and largest negative integer that are upper and lower bounds , respectively for the real zeros of (B ) Appr

Here's a problem somebody posted 5 years ago in 2020

1167931 (2020-10-22 08:58:07):

Use the upper and lower bound theorem find the smallest positive integer and
largest negative integer that are upper and lower bounds , respectively for the
real zeros of



Notice the rows of numbers at the bottom of the long divisions by
 -3 through 5:

-3|1 -1 -8 -12 -25
  |  -3 12 -12  72
   1 -4  4 -24  47  <-- notice that the signs on the bottom row of the synthetic
                        division DO INDEED alternate.  That means that -3 IS
                        INDEED a lower bound for negative zeros.

-2|1 -1 -8 -12 -25
  |  -2  6   4  16
   1 -3 -2  -8  -9  <-- notice that the signs on the bottom row of
                        the synthetic division DO NOT alternate.  That means
                        that -2 IS NOT a lower bound for negative zeros. 

-1|1 -1 -8 -12 -25
  |  -1  2   6   6
   1 -2 -6  -6 -19  <-- notice that the signs on the bottom row of
                        the synthetic division DO NOT alternate.  That means
                        that -1 IS NOT a lower bound for negative zeros.

 0|1 -1 -8 -12 -25
  |   0  0   0   0
   1 -1 -8 -12 -25  <-- notice that the signs on the bottom row of
                        the synthetic division DO NOT alternate nor are they all
                        the same.  That means that 0 IS NOT a lower bound for
                        negative zeros nor an upper bound for positive zeros.

 1|1 -1 -8 -12 -25
  |   1  0  -8 -20
   1  0 -8 -20 -45  <-- notice that the signs on the bottom row of
                        the synthetic division are NOT all the same. That means
                        that 1 IS NOT an upper bound for positive zeros. 

 2|1 -1 -8 -12 -25
  |   2  2 -12 -48
   1  1 -6 -24 -73  <-- notice that the signs on the bottom row of
                        the synthetic division are NOT all the same.  That means
                        that 2 IS NOT an upper bound for positive zeros. 

 3|1 -1 -8 -12 -25
  |   3  6  -6 -54
   1  2 -2 -18 -79  <-- notice that the signs on the bottom row of
                        the synthetic division are NOT all the
                        same.  That means that 3 IS NOT an upper bound for
                        positive zeros. 

 4|1 -1 -8 -12 -25
  |   4 12  16  16
   1  3  4   4  -9  <-- notice that the signs on the bottom row of
                        the synthetic division are NOT all the same.  That means
                        that 4 IS NOT an upper bound for positive zeros.

 5|1 -1 -8 -12 -25
  |   5 20  60 240
   1  4 12  48 215  <-- notice that the signs on the bottom row of
                        the synthetic division are ALL THE SAME. That means that
                        5 IS INDEED an upper bound for positive zeros.

I don't know if this upper and lower bound theorem for roots or zeros of
polynomials is taught anymore.  Apparently it still was back in 2020.

Anyway the largest negative integer that is a lower bound for negative zeros is
-3 and the smallest positive integer that is an upper bound for positive zeros is 5.

I'm not going to do part (B) for it's long and involved and done with synthetic
division.  I just wondered if they bother teaching these long methods anymore,
since calculators and computers can find them so quickly.

Ikleyn complains about the AI "tutor", but in 10 years it'll be able to solve
anything humans can solve.

Edwin

>In his post,  Edwin  AGAIN  makes lunge toward me,  without any visible reason/cause from my side.<<

I made no lunge at you, Ikleyn, where did you get that? I merely said that you
complain about the AI "tutor", which you do because it has not learned
everything in mathematics yet.  Then I said "but in 10 years it'll be able to
solve anything humans can solve."  And it will be, and will take over many jobs.
You recently referred to something AI had written as "gibberish".  Is that not complaining?

>>>Edwin,  what  I  do for this  AI  by checking their errors,
and what  I  do for visitors of this forum,  explaining/revealing the errors of
the  AI,  is  INVALUABLE  SERVICE  for them.  But you either do not understand
it  (unfortunately), or deliberately try to throw black paint on what  I  do at
this forum.
Edwin, let me tell you how an intelligent gentleman would review my work:
    "Thanks to tutor @ikleyn for her tireless daily work on finding,  identifying and fixing
    the errors made by artificial intelligence.
    This work brings closer the day when artificial intelligence will become a reliable assistant
    to teachers and students in their daily work."
It is a true intelligent form and a true intelligent style of writing a review,
and it is a style of how a true professor should write.
I decided to share this  " HOW TO "  with you,  in case if you are unfamiliar with it.

Also, @ikleyn does not complain about  Artificial  Intelligence
(because ikleyn is not an idiot to complain about  AI).

Are you denying recently calling the AI's output "gibberish"?

@ikleyn makes hidden errors,  made by the  AI,  visible and fixes them,  if to tell truth.

Edwin,  English is your native language,  isn't it ?   And you are a professor,  right ?

Yes, I was a mathematics prof for 40 years, from 1966 through 2005. I am now 88
years of age.

Then why it is so difficult for you to tell truth in its proper and appropriate form,  using right terms ?

What falsehood do you think I told? I said that in 10 years AI will be able to
solve anything.  I stand by that.

Edwin