elementary number theory - Why does the rational root theorem …
2015年3月20日 · $\begingroup$ The theorem refers to the numerator and denominator of a possible rational root, saying these divide the constant term and leading term. If you allow noninteger coefficients, at least the constant term and lead term would have to be integers, or it wouldn't make sense to look for numerator and denominator being divisors of them.
Proof for rational roots - Mathematics Stack Exchange
Note that this theorem is called the Rational Root Theorem! Part B: Roots of $9x^3+18x^2-4x-8=0$ By the Rational Root Theorem, we have the possible roots as$$\begin{align*} & \pm1\pm2\pm4\pm8\\ & \pm1\pm3\\ & \implies\pm\frac 13,\pm\frac 23,\pm\frac 43,\pm\frac 83,\pm1,\pm2,\pm4,\pm8\end{align*}\tag4$$ Testing out the points, we find that $-2 ...
Proving Gauss' polynomial theorem (Rational Root Test)
Proving a statement involving the rational root theorem. 0. Two recurrence relations gcd proof. 2.
Why doesn't the Rational Root Theorem work on $x^3-6x^2+4x …
When you apply the rational root theorem, you find all the rational roots, if there are any. If the theorem finds no roots, the polynomial has no rational roots. (For a cubic, we would observe that the polynomial is irreducible over the rationals.
Concerns with the proof of the rational root theorem
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polynomials - Rational root theorem - Mathematics Stack Exchange
Without such fundamental results the Rational Root Test need not hold, e.g. it may fail in rings of algebraic integers. $\endgroup$ – Bill Dubuque Commented Aug 25, 2016 at 21:22
Does the Rational Root Theorem ever guarantee that a polynomial …
Using the rational root theorem you can tell if a given polynomial with integer coefficients has rational roots. If the degree of the polynomial is greater than $3$ this theorem tells you nothing. For instance consider $(x^2-2)(x^2+2)=x^4-4$ which doesn't have rational roots, but …
algebra precalculus - Two Questions On The Rational Root …
2018年8月2日 · Note that the Rational Root Theorem in no way guarantees the existence of rational roots. It does, however, give information about the nature of rational roots should they exist. There are some instances where the Rational Root Theorem is sufficient to find all the real roots of a polynomial.
What is the Rational Root Theorem? : r/learnmath - Reddit
The rational root theorem says that the rational roots of a polynomial with integer coefficients have the form of a factor of the constant term divided by a factor of the leading coefficient; this is useful for solving polynomial equations, because it allows you to focus your attention on a few possible linear factors with integer coefficients ...
Problem with the rational root theorem - Mathematics Stack …
You don't have all integer coefficients of your middle terms in the expanded polynomial. Multiplying through by three will get rid of fractional coefficients and give you the possibility of 3 being a root. The rational root theorem requires all integer coefficients.