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Math Help - GRE subject test question

  1. #1
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    GRE subject test question

    Hello all,

    Here is a sample GRE subject exam question.

    Let p(x) be the polynomial x^3 + a*x^2 + b*x + c, where a, b, and c are real constants. if p(-3) = p(2) = 0 and p'(3) < 0, which of the following is a possible value of c?
    (A) -27
    (B) -18
    (C) -6
    (D) -3
    (E) -1/2

    I have figured out the solution getting -27 as the answer. My question regards a solution I came across on another forum by someone else that I am having trouble following.

    Given the information available is it possible to conclude that the third root of the equation must be strictly less than -3?

    Thanks!
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by kkoutsothodoros View Post
    Hello all,

    Here is a sample GRE subject exam question.

    Let p(x) be the polynomial x^3 + a*x^2 + b*x + c, where a, b, and c are real constants. if p(-3) = p(2) = 0 and p'(3) < 0, which of the following is a possible value of c?
    (A) -27
    (B) -18
    (C) -6
    (D) -3
    (E) -1/2

    I have figured out the solution getting -27 as the answer. My question regards a solution I came across on another forum by someone else that I am having trouble following.

    Given the information available is it possible to conclude that the third root of the equation must be strictly less than -3?

    Thanks!
    There is something wrong with the question, since:

    p(x) = (x+3)(x-2)(x-r3)

    p'(x) = 3 x^2 + 2x (1 - r3) - r3 - 6

    Then p'(3)=27 - 7 r3<0, which implies that r3>0.

    RonL
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  3. #3
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    p(x) = (x + 3)(x - 2)(x - r3) = x^3 + (1 - r3)*x^2 - (6 + r3)*x + 6*r3.

    so c = 6*r3

    p'(-3) = 27 - 6*(1 - r3) + (6 + r3) = 27 - 6 + 6*r3 + 6 + r3 = 27 + 7*r3 < 0.

    so -7*r3 > 27 or r3 < -27/7.

    So c < -18 leaving -27 to be the only possible answer.

    My question is if it's possible to conclude that r3 is < -3 right off the bat without doing any of thw work above.
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