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Rational Zero Theorem


If the coefficients of the polynomial

 d_nx^n+d_(n-1)x^(n-1)+...+d_0=0
(1)

are specified to be integers, then rational roots must have a numerator which is a factor of d_0 and a denominator which is a factor of d_n (with either sign possible). This follows since a polynomial of polynomial order n with k rational roots can be expressed as

 (a_1x+b_1)(a_2x+b_2)...(a_kx+b_k)(c_(n-k)x^(n-k)+...+c_0)=0,
(2)

where the roots are x_1=-b_1/a_1, x_2=-b_2/a_2, ..., and x_k=-b_k/a_k. Factoring out the a_is,

 a_1a_2...a_k(x+(b_1)/(a_1))(x+(b_2)/(a_2))...(x+(b_k)/(a_k))(c_(n-k)x^(n-k)+...+c_0)=0.
(3)

Now, multiplying through,

 a_1a_2...a_kc_(n-k)x^n+...+b_1b_2...b_kc_0=0,
(4)

where we have not bothered with the other terms. Since the first and last coefficients are d_n and d_0, all the rational roots of equation (1) are of the form [factors of d_0]/[factors of d_n].


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References

Bold, B. Famous Problems of Geometry and How to Solve Them. New York: Dover, p. 34, 1982.Niven, I. M. Numbers: Rational and Irrational. New York: Random House, 1961.

Referenced on Wolfram|Alpha

Rational Zero Theorem

Cite this as:

Weisstein, Eric W. "Rational Zero Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RationalZeroTheorem.html

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