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opus

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## Main Question or Discussion Point

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

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.