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The Rational Roots Test: Examples (page 2 of 2)

  • Find all possible rational x-intercepts of y = 2x3 + 3x 5.

    Keeping in mind that x-intercepts are zeroes, I will use the Rational Roots Test.

    The constant term of this polynomial is 5, with factors 1 and 5.

    The leading coefficient is 2, with factors 1 and 2.

    Then the Rational Roots Tests yields the following possible solutions:


    x =  1, 1/2, 5, 5/2

Don't forget the "plus-or-minus" on the solution. You either need to list out all the possible solutions separately, as I did in the first example; or use a "plus-or-minus" in front of each possible solution, as I showed here; or put one "plus-or-minus" in front of the whole list of possible solutions. as I will show in the next example. Just make sure you have a "plus-or-minus" in there somewhere.


By the way, as the graph shows, if there is a rational root for y = 2x3 + 3x 5, it has to be at x = 1.


graph of y = 2x^3 + 3x - 5

  • Use the Rational Roots Test to find all possible rational zeroes of
    6x4 11x3 + 8x2 33x 30.

    This problem will be more complicated than the previous one, because the leading coefficient is not a simple "1". Copyright Elizabeth Stapel 2002-2011 All Rights Reserved

    The constant term is 30, with factors 1, 2, 3, 5, 6, 10, 15, and 30.

    The leading coefficient is 6, with factors 1, 2, 3, and 6.

    Then the Rational Roots Test yields:

      list of possible roots

Yes, this is a very long list. Warning: You should expect at least some of your homework exercises and at least one test question to be as long as this.

Check out the graph:



You can see from the graph that there may be rational roots at x = 2/3 and x = + 5/2, but it would probably not make sense to try any of the other listed potential zeroes.

In those last two examples, please note how I was orderly in listing out the fractions, taking the time to reduce each fraction and to discard duplicates from the list. Take the time to work in the same orderly fashion, because this really is a simple topic, and it would be a shame if you lost points on the test only because you were careless.

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Cite this article as:

Stapel, Elizabeth. "The Rational Roots Test: Examples." Purplemath. Available from Accessed


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