How do you use the rational zero theorem to list all possible rational zeros for the given function #f(x)=x^3-14x^2+13x-14#?

1 Answer
Feb 18, 2015

Hello,

Let #P = a_n X^n + a_{n-1}X^{n-1} + \ldots + a_1X+a_0# a polynom with #a_0,\ldots, a_n# integers. Suppose that #a_n\ne 0# and #a_0\ne 0#.

If irreductible fraction #\frac{p}{q}# is a root of #P#, then #p# is a factor of #a_0# and #q# is a factor of #a_n#.

In your example, the factors of #a_0=-14# are
#-1,1,-2,2-7,7,-14,14#,
and the factors of #a_n=1# are just #-1,1#.

Therefore, you have some candidates for the rational roots :
#-1,1,-2,2-7,7,-14,14#.

Remark. If #a_n=1#, the rational roots are necessarily integers !

You can check that no one is solution !