# use the rational zeros theorem to find all the real zeros of the polynomial function f(x)=3x^3-x^2+3x-1  must show work , simplify use radicals as needed use integers or fractions

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embizze | High School Teacher | (Level 1) Educator Emeritus

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Find the real zeros of `f(x)=3x^3-x^2+3x-1` using the rational root threorem:

By the rational root theorem the possible rational roots are `+-1,+-1/3`

(The rational roots are of the form `p/q` where p is a factor of the constant term and q a factor of the leading coefficient)

You can use polynomial long division or synthetic division to check each.

We find that `1/3` is a zero, and we have `x^3-x^2+3x-1=(x-1/3)(3x^2+3)`

`3x^2+3=3(x^2+1)` which has no real zeros. (The sum of squares does not factor over the reals -- or you could use the quadratic formula to find the discriminant is less than 0)

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The real zero of `f(x)=3x^3-x^2+3x-1` is `x=1/3`

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The graph:

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