You can also use the rational root theorem: "If the rational number

is a root of the polynomial

, then n evenly divides the leading coefficient,

, and m evenly divides the constant term,

."

Of course, there is no guarantee that a polynomial equation

**has** a rational root but if it does, that limits the possible roots. Here, your equation is [tex]x^3- x^2+ 12x+ 32= 0[tex]. The leading coefficient is 1 so the denominator of any rational root must evenly divide 1- i.e. must be

so any rational root must be an integer. The constant term is 32 so if there is a rational root it must be

, [tex]\pm 2[tex],

,

,

, or

. Try those and see if any work.