Question

Using complete sentences, explain how the outcome of the Rational Root Theorem and Descartes’ Rule of Signs differ from one another. (2 points)

Answer 1

Couldn't you just say that Descartes' rule of signs will tell you (or help narrow down) the NUMBER of positive real zeros (but not what they ARE), and the Rational Zero Theorem will list every possible factor in the constant & leading coefficient, of which there is some combination that will be every rational zero (and a lot of junk left over).

Answer 2

the Rational Root Theorem gives you information about how many rational roots there could be. Descartes' Rule of Signs gives you information about how many positive or negative roots there could be (which may or may not be rational). for example, [ if p(x) = x^2 + 1, the rational root test tells us that the only rational roots (if they exist) will be -1, and 1. applying the rule of signs, we find that there are no positive roots (because there are no sign changes for p(x)), and there are no negative roots (because there are no sign changes for p(-x), which happens to equal p(x) in this case). so as you can see, the two tests can yield different results.

Answer 3

Rational Root Theorem tells you the possible rational roots.
if \(p(x)=a_nx^n+...+a_0\) then IF it has a rational root \(\frac{p}{q}\) the numerator \(p\) must divide the constant \(a_0\) and the denominator \(q\) must divide the leading coefficient \(a_n\)
but this is really an historical relic. no one says the roots are rational for one thing.
for another it says nothing about which ones are actual roots.
in the modern age one uses a computer algebra system to find the roots

Answer 4

Thanks so much you guys!