Possible Rational Roots

Problem 261: Given the polynomial

$y = 10x^3 - 57x^2 + 36x - 5$

list the possible rational roots.

Solution: 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 -5 are: -1, 5, 1, -5.

The possible roots are given by

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

Simplifying, the possible roots are

$\frac{p}{q} = \pm 1, \pm \frac{1}{2}, \pm \frac{1}{5}, \pm \frac{1}{10}, \pm 5, \pm \frac{5}{2}.$

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