Possible Rational Roots II

Problem 262: Given the polynomial

$y = 41x-21x^2 - 10,$

list the possible rational roots.

Solution: First, we need to rearrange the given polynomial from highest to smallest.

$y = 41x - 21x^2 - 10 = -21x^2 + 41x - 10.$

To find the possible roots of this polynomial we have to consider the possible factors of leading coefficient (call it $q$ ) and the constant coefficient (call it $p$ ).

That is, the possible factors of 10 are: -10, -1, 1, 10, -2, -5, 2, 5.

The possible factors of -21 are: -1, 21, 1, -21, -7, 3,-3, 7.

The possible roots are given by

$\frac{p}{q} = \frac{-10, -1, 1, 10, -2, -5, 2, 5}{-1, 21, 1, -21, -7, 3,-3, 7} = \frac{\pm 1, \pm 2, \pm 5, \pm 10}{\pm 1, \pm 7, \pm 21, \pm 3}.$

Simplifying, the possible roots are

$\begin{align*} \frac{p}{q} & = \pm 1, \pm \frac{1}{7}, \pm \frac{1}{21}, \pm 2, \pm \frac{2}{7}, \pm \frac{2}{21}, \\ & \pm 5, \pm \frac{5}{7}, \pm \frac{5}{21}, \pm 10, \pm \frac{10}{7}, \pm \frac{10}{21}, \\ & \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{5}{3}, \pm \frac{10}{3}.\end{align}$

Leave a Reply Cancel reply