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Math Help - Solving for x: polynomial equation

  1. #1
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    Solving for x: polynomial equation

    (I really hope this is in the correct place.)

    The equation is 3x^4 + 8x^3 - 30x^2 -72x +27 I need to find the four roots but I only managed three, and one was by looking and guessing. In an exam, I won't have time to start guessing so I was wondering if there was a more methodic, secure way to solve this.

    I guessed (using the calculator) that 3 was one root. Through that, I guessed -3. Then I divided the equation by (x^2 - 9) and got  3x^2 + 8x - 3 which, in turn, gave me x=\frac{1}{3} and x=-3. But that's what I don't understand, I divided the original equation by (x+3), how am I still getting that as a result? And since the highest power is 4, I assumed that I would find four roots but I've only found three.

    Also, once I have all four answers, the question stated that I need to write them out in order, from the most amount of times that the root appears in the equation. I don't exactly understand what this means. I would post the original question, but it's not in English and therefore wouldn't really be of any value. I translated as best as I could.

    Thank you.
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  2. #2
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    Quote Originally Posted by dantap View Post
    (I really hope this is in the correct place.)

    The equation is 3x^4 + 8x^3 - 30x^2 -72x +27 I need to find the four roots but I only managed three, and one was by looking and guessing. In an exam, I won't have time to start guessing so I was wondering if there was a more methodic, secure way to solve this.

    I guessed (using the calculator) that 3 was one root. Through that, I guessed -3. Then I divided the equation by (x^2 - 9) and got  3x^2 + 8x - 3 which, in turn, gave me x=\frac{1}{3} and x=-3. But that's what I don't understand, I divided the original equation by (x+3), how am I still getting that as a result? And since the highest power is 4, I assumed that I would find four roots but I've only found three.

    Also, once I have all four answers, the question stated that I need to write them out in order, from the most amount of times that the root appears in the equation. I don't exactly understand what this means. I would post the original question, but it's not in English and therefore wouldn't really be of any value. I translated as best as I could.

    Thank you.

    Assuming you did the calculations correctly it means that -3 is a double root, so the roots are -3, -3, 1/3, 3.

    Tonio
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  3. #3
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    The easiest way is to check the factors of 27. Any linear factors will be of the form x - a where a is a factor of 27.
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  4. #4
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    You have to guess the first root to find the other.For this root you have to search for the numbers wich divide 27.These are +1,-1,+3,-3,+9,-9.
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