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Concept
Rational root theorem
Summary
In algebra, the rational root theorem (or rational root test, rational zero theorem, rational zero test or p/q theorem) states a constraint on rational solutions of a polynomial equation
with integer coefficients and . Solutions of the equation are also called roots or zeros of the polynomial on the left side.
The theorem states that each rational solution x = p⁄q, written in lowest terms so that p and q are relatively prime, satisfies:
p is an integer factor of the constant term a0, and
q is an integer factor of the leading coefficient an.
The rational root theorem is a special case (for a single linear factor) of Gauss's lemma on the factorization of polynomials. The integral root theorem is the special case of the rational root theorem when the leading coefficient is an = 1.
The theorem is used to find all rational roots of a polynomial, if any. It gives a finite number of possible fractions which can be checked to see if they are roots. If a rational root x = r is found, a linear polynomial (x – r) can be factored out of the polynomial using polynomial long division, resulting in a polynomial of lower degree whose roots are also roots of the original polynomial.
The general cubic equation
with integer coefficients has three solutions in the complex plane. If the rational root test finds no rational solutions, then the only way to express the solutions algebraically uses cube roots. But if the test finds a rational solution r, then factoring out (x – r) leaves a quadratic polynomial whose two roots, found with the quadratic formula, are the remaining two roots of the cubic, avoiding cube roots.
Let with
Suppose P(p/q) = 0 for some coprime p, q ∈ Z:
To clear denominators, multiply both sides by qn:
Shifting the a0 term to the right side and factoring out p on the left side produces:
Thus, p divides a0qn. But p is coprime to q and therefore to qn, so by Euclid's lemma p must divide the remaining factor a0.
On the other hand, shifting the an term to the right side and factoring out q on the left side produces:
Reasoning as before, it follows that q divides an.
Official source
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