Rational Root Theorem

Algebra 2 · Axiom Academy

LESSON The Rational Root Theorem Every rational zero of an integer-coefficient polynomial has the form — where p divides the constant and q divides the leading coefficient. Write the polynomial with integer coefficients . The theorem chains a rational zero's two parts to two specific coefficients: its numerator to the constant term, its denominator to the leading coefficient. p is a factor of the constant term a_0 , q is a factor of the leading coefficient a_n . The general integer-coefficient polynomial The two divisibility conditions a rational zero must satisfy Substitute into the polynomial and clear the denominators by multiplying through by q^n . Watch what happens to the factor q : every term except the first carries at least one q . That single observation is the whole proof. Multiply every term by q^n to clear fractions Isolate the one term without a q The right side is a multiple of q , so q divides a_n p^n . Because , q shares no factor with p^n — so q must divide a_n . Grouping instead by p gives in the same way. 3. Building the Candidate List Apply the theorem to f(x) = 2x^3 - 5x^2 - 4x + 3 . The constant term is a_0 = 3 and the leading coefficient is a_n = 2 . Feed every over every through the factory, reduce, and drop duplicates. Numerators from the constant, denominators from the leading coefficient The complete list of possible rational zeros — eight candidates

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