Is there a faster way to find the roots of a polynomial using Horners method?

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Discussion Overview

The discussion revolves around finding the roots of the polynomial $$x^3-8x^2+19x-12$$, specifically exploring whether there is a faster method than Horner's method. Participants consider various approaches, including the Rational Root Theorem and the Newton-Raphson method, while debating their efficiency.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses familiarity with Horner's method but seeks a quicker alternative for finding roots.
  • Another suggests using the Rational Root Theorem to factor the polynomial over integers.
  • A participant mentions the Newton-Raphson method as a potential way to find at least one real root for the cubic polynomial.
  • Some participants question the efficiency of the Rational Root Theorem compared to Horner's method, suggesting it may not be quicker.
  • There is a discussion about the computational complexity of Horner's method being $\mathcal{O}(n)$, while opinions vary on the efficiency of the Rational Root Theorem in practice.
  • One participant notes that if the polynomial does not have rational roots, the Rational Root Theorem may fail to provide a solution.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether the Rational Root Theorem is a faster method than Horner's method. There are competing views regarding the efficiency of both methods, and the discussion remains unresolved.

Contextual Notes

Some participants express uncertainty about the application of the Rational Root Theorem and its effectiveness, particularly in cases where rational roots may not exist.

theakdad
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I have given polynomial: $$x^3-8x^2+19x-12$$

I know how to find the roots with Horners method,i am just wondering if there is an easier and quicker way to find them? Thank you!
 
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wishmaster said:
I know how to find the roots with Horners method

You don't have to go that far. Try proving that this factors over integers completely by applying rational root theorem (integral root theorem for monic polynomials).
 
mathbalarka said:
You don't have to go that far. Try proving that this factors over integers completely by applying rational root theorem (integral root theorem for monic polynomials).

I don't know how to do it...
 
wishmaster said:
I have given polynomial: $$x^3-8x^2+19x-12$$

I know how to find the roots with Horners method,i am just wondering if there is an easier and quicker way to find them? Thank you!

The polynomial has degree 3, so that it exists at least one real root that can be found with the Newton-Raphson method...
 
Find the divisors of $$12$$ then divide that by the divisors of the leading term $$1$$ so we have $$\pm 1 , \pm 3 , \pm 4 , \pm 6 \pm 12$$ as possible roots.

$$P(3) = 3^3-8(9)+19\times 3-12 = 27-72+57-12=0 $$

Hence $$x=3$$ is a solution . The next step divide the cubic by $$(x-3)$$ to get a polynomial of degree $2$.
 
Seems that's not the easier and faster method as Horners...or i can't see it! :D
 
wishmaster said:
Seems that's not the easier and faster method as Horners...or i can't see it! :D

... not only neither easier nor faster... if the polynomial doesn't have rational roots [... the most probable case...] it fails!(Tmi)...

Kind regards

$\chi$ $\sigma$
 
  • #10
chisigma said:
... not only neither easier nor faster... if the polynomial doesn't have rational roots [... the most probable case...] it fails!(Tmi)...

Kind regards

$\chi$ $\sigma$

My question was about rational roots...how to find them fast. (Mmm)
 
  • #11
chisigma said:
neither easier nor faster

Horner's method has computational complexity $\mathcal{O}(n)$, whereas RRT exceeds that quite less often, as far as I can tell.
 
  • #12
mathbalarka said:
Horner's method has computational complexity $\mathcal{O}(n)$, whereas RRT exceeds that quite less often, as far as I can tell.

So i think the answer is: Deal with it as you can?
 
  • #13
Nope. I believe RRT is much faster at several cases than Horner, and that's what I answered above using computational complexity theory.
 

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