How do you use the rational root theorem to list all possible roots for #12x^4+14x^3-5x^2-14x-4=0#?

1 Answer
George C.
Aug 30, 2017

The only possible rational roots are:

#+-1/12, +-1/6, +-1/4, +-1/3, +-1/2, +-2/3, +-1, +-4/3, +-2, +-4#

...but none of these is a root.

So this equation has no rational roots.

Explanation:

Given:

#12x^4+14x^3-5x^2-14x-4 = 0#

By the rational root theorem, any rational roots must be expressible in the form #p/q# for integers #p, q# with #p# a divisor of the constant term #-4# and #q# a divisor of the coefficient #12# of the leading term.

So the only possible rational roots are:

#+-1/12, +-1/6, +-1/4, +-1/3, +-1/2, +-2/3, +-1, +-4/3, +-2, +-4#

In practice, none of these is a root, so this quartic equation has no rational roots.

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