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Definition of rational root theorem

The Rational Root Theorem

The Rational Root Theorem says if a polynomial equation a_n x^n + a_{n - 1} x^{n - 1} + ... + a_1 x + a_0 = 0 has rational root \frac{p}{q} (p, q \in \mathbb{Z}) then the denominator q divides the leading coefficient and the numerator p divides a_0.

As an addition to this theorem, for every whole number k, number p – kq is a divisor of f(k).

the Rational Root Theorem

Example 1. Find all rational roots of the following equation:

The leading coefficient is 5 which means that, since q divides it, is from the set {-1, 1, -5, 5} and the free coefficient is number 3 which means that p is from the set {-1, 1, -3, 3}.

Since we know possibilities for q and p, we can find all combinations to see what our solutions, \frac{p}{q} can be.

\frac{p}{q} \in \{-1, 1, -\frac{1}{5}, \frac{1}{5}, -3, 3, -\frac{3}{5}, \frac{3}{5} \}

The first solution is 1. When we factorize given equation we get:

5x^3 - 7x^2 - x + 3 = (x - 1)(5x^2 - 2x - 3)

Since the other factor is a quadratic polynomial, we can easily find its roots. As the final result we get:

x_1 = 1, x_2 = 1, x_3 = -\frac{3}{5}