Possible upper and lower bounds definition typo for real zeros of polynomial functions.

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Zak
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Possible upper and lower bounds definition typo for real zeros of polynomial functions.

Zak
Shouldn't the signs of the final line of the division tableau be all positive for c > 0 to make it the upper bound? I don't think that merely having the same signs would suffice. They must all be positive, unless I am mistaken. My other textbooks state that they must all be positive for this upper bound. I did notice this effect on a practice problem as well where all signs were negative for the final line but it wasn't an upper bound. Problem #16 of the 3rd corrected edition. 4 gives all negatives but is not an upper bound. Thanks.
Zak
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Re: Possible upper and lower bounds definition typo for real zeros of polynomial functions.

Zak
I made a mistake on problem 16. The final line isn't all negative signs.
Zak
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Re: Possible upper and lower bounds definition typo for real zeros of polynomial functions.

Zak
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Perhaps the upper bounds holds when all are negative? Stewart's Precalculus mentions all positive so I would be interested in this.
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Re: Possible upper and lower bounds definition typo for real zeros of polynomial functions.

Carl Stitz
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Hello!

Great question!

So let me think out loud for a minute.

So the idea is that when you divide by x=c, your polynomial is:

p(x) = (x-c)(q(x)) + r

Let's say the signs of the division tableau are all (-) and I pick k > c > 0.

Then p(k) = (k-c)(q(k)) + r

Now (k-c) > 0 so (k-c) is (+)

q(k) is a sum of terms which are powers of k (all (+)) times negative coefficients.  So that means q(k) is (-).

Finally, r is (-) as well.

So p(k) = (+)(-) + (-) = (-) which means k can't be a zero.

Let know what you think!

Carl
Zak
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Re: Possible upper and lower bounds definition typo for real zeros of polynomial functions.

Zak
Thanks!

Let me look at your example, and I'll post back.