# use the rational zero theorem to find all the real zeros of polynomial, use the zeros to factor f over the real numbers f(x)=11x^3-x^2+11x-1 x= f(x) =show work

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The rational zero theorem says that if the zeros of the polynomial `f(x)` are rational, then numerators are positive or negative factors of the constant term -1, and the denominators are positive factors of the leading coefficient 11.

This means we can look at any of the possible roots:

`+-1` , `+-1/11`

Starting with `x=1/11` , we see that

`f(1)=1/121-1/121+1-1=0` so `x-11` is a factor.

With factoring, we see then that

`f(x)=11x^3-x^2+11x-1`

`=x^2(11x-1)+1(11x-1)`

`=(11x-1)(x^2+1)`

**So the only real root is `x=1/11` .**

**Sources:**