Question

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

Answer 1

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`.