text: # 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
  
  embizze | High School Teacher | (Level 1) Educator Emeritus
  
  Posted on
  
  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)
  
  --------------------------------------------------------------------
  
  The real zero of `f(x)=3x^3-x^2+3x-1` is `x=1/3`
  
  --------------------------------------------------------------------
  
  The graph:
explanation: The document contains a math question and a detailed answer to the question. The question asks for finding the real zeros of a polynomial function using the rational zeros theorem, and the answer provides a step-by-step solution.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # 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
  
  embizze | High School Teacher | (Level 1) Educator Emeritus
  
  Posted on
  
  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)
  
  --------------------------------------------------------------------
  
  The real zero of `f(x)=3x^3-x^2+3x-1` is `x=1/3`
  
  --------------------------------------------------------------------
  
  The graph:
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and a detailed answer to the question. The question asks for finding the real zeros of a polynomial function using the rational zeros theorem, and the answer provides a step-by-step solution.
  
  Label: question-and-answer