How do you use the rational root theorem to find the roots of #3x^3 - kx^2 - 16x + 12 = 0#?

1 Answer
Jul 31, 2015

Answer:

Without knowing the value of #k# we can not determine which of the candidate roots identified by the rational root theorem are valid.

Explanation:

The rational root theorem identifies all candidate rational polynomial roots (I found 18 candidates for this example) but without knowing the value of #k# we can not determine which of these are valid.

[Perhaps #k# was a typing error or perhaps the question might have been for what values of #k# does the given expression have integer roots?]

Rational Root Theorem
#color(white)("XXXX")##sum_(i=0)^n a_ix^i = 0#
every rational solution is within the set
#color(white)("XXXX")##+-("factors of "a_0)/("factors of "a_n)#

For the given equation the candidates for rational solutions are
#color(white)("XXXX")##+- ({1,2,3,4,6,12})/({1,3})#

#color(white)("XXXX")##= {+-1, +-2, +-3, +-4, +-6, +-12, +-1/3, +-2/3, +-4/3}#

The only way to determine which of these candidates are valid is by evaluating the polynomial for each candidate value to see if the result is #0#.