rational roots

[kangsang24]

I dont know why I cant figure this out. 0=x^3-3x+5 using rational roots P/Q I get (p)5: 1, 5 (q)1: 1 (p/q): +/- 1, 1/5, and 5. None of those are zeros but I know there is an x intercept around -2.7. What am I missing????


After reviewing the problem, I realize that there are no rational roots; however, there still is a root (maybe im using the wrong word here, is root interchangeable for x-intercept?) for "x" in the equation "y=x^3-3x+5" I can see through the graphing calculator that there is an x-intercept around -2.7. How can I find this algebraically??

 

[Bob Brown MSEE]

No rational roots exist

0=x^3-3x+5

 

[kangsang24]

how do I find the zero without using a calculator... and there is a zero. Again its around -2.7. Graph it and youll see. Shouldnt rational roots theorem find this root?

 

[stapel]

I dont know why I cant figure this out. 0=x^3-3x+5 using rational roots P/Q I get (p)5: 1, 5 (q)1: 1 (p/q): +/- 1, 1/5, and 5. None of those are zeros but I know there is an x intercept around -2.7. What am I missing????

What, exactly, were the instructions?

I suspect that the test-writer is aware of graphing calculators, so he didn't want to ask a question about the zeroes of a function for which the student could cheat by just looking at the graph. He wanted to see if the student knew how to find the list of possible rational roots, so he gave an equation with did not have rational roots. The question, I suspect, actually asked you to find the possible roots.

P.S. Check the formula for this. I think you've made a slight error in your computations.

 

[Quaid]

how do I find the zero without using a calculator

Hi Kangsang24:

You may use the cubic formula, to determine the exact zero. (This is not "easy".)

You may use a numerical approach (such as Newton's Method or guess-and-check), to approximate the zero. (Newton's Method requires some knowledge of differential calculus.)

there is a zero. Again its around -2.7. Graph it and youll see.

I don't see that, on the graph. I see a zero much closer to -2 than -3.

Shouldnt rational roots theorem find this root?

No.

The Rational Roots Theorem does not find any root. It merely gives you a list of possible candidates for rational roots. YOU then determine any rational root(s), by checking (or eliminating) the candidates.


:idea: If a root is not rational, then it will not appear in a list of candidates coming from the Rational Root Theorem.


Did you read Bob Brown's reply? This root is NOT A RATIONAL NUMBER.


By the way, here is the EXACT root, as determined by the cubic formula. (Bob Brown also posted this number, albeit expressed differently.)

x = -1/2*(20+4*sqrt(21))^(1/3)-2/(20+4*sqrt(21))^(1/3)

Cheers :cool: