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

Jul 31, 2015

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
$\textcolor{w h i t e}{\text{XXXX}}$${\sum}_{i = 0}^{n} {a}_{i} {x}^{i} = 0$
every rational solution is within the set
$\textcolor{w h i t e}{\text{XXXX}}$$\pm \left(\text{factors of "a_0)/("factors of } {a}_{n}\right)$

For the given equation the candidates for rational solutions are
$\textcolor{w h i t e}{\text{XXXX}}$$\pm \frac{\left\{1 , 2 , 3 , 4 , 6 , 12\right\}}{\left\{1 , 3\right\}}$

$\textcolor{w h i t e}{\text{XXXX}}$$= \left\{\pm 1 , \pm 2 , \pm 3 , \pm 4 , \pm 6 , \pm 12 , \pm \frac{1}{3} , \pm \frac{2}{3} , \pm \frac{4}{3}\right\}$

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 $0$.