# 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)

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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:
The text passage contains a mathematical question about finding the real zeros of a polynomial function using the rational zeros theorem, and it provides a detailed solution to the problem.
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Given a text passage, determine:
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    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
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- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
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# 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:
user
The text passage contains a mathematical question about finding the real zeros of a polynomial function using the rational zeros theorem, and it provides a detailed solution to the problem.

TYPE: SINGLE_QUESTION_WITH_ANSWER