Explore: Rational Root Theorem Solutions

Rational Root Theorem Solutions

The Rational Root Theorem (RRT) is used to determine the possible rational roots (zeros, x-intercepts) for a polynomial containing only integer coefficients .
Potential rational roots, $x = \frac{Factors of the constant}{Factors of the leading coefficient}$
Use ± with all factors because rational numbers can be positive or negative.
Recall that a rational number is a number that can be written in the form $\frac{p}{q}$ in which p and q are integers and q is not equal to zero .

Example 5

List all potential rational roots.

$2 x^{3} - 9 x^{2} + 7 x + 6 = 0$

Factors of the constant: $\pm 1, \pm 2, \pm 3, \pm 6$

Factors of the leading coefficient: $\pm 1, \pm 2$

There are 12 possible rational roots.

$\begin{array}{rcl} RRT & = & \frac{\pm 1, \pm 2, \pm 3, \pm 6}{\pm 1, \pm 2} \\ = & \pm 1, \pm \frac{1}{2}, \pm 2, \pm 3, \pm \frac{3}{2}, \pm 6 \end{array}$