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Math Help - Can't figure out how.. (Pre-Calc, Rational Zero Theorem and Synthetic Division

  1. #1
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    Can't figure out how.. (Pre-Calc, Rational Zero Theorem and Synthetic Division

    I've encountered the problem:

    x^3 - 10x - 12

    I am suppose to;
    a.) list all possible rational roots
    b.) use synthetic division to test
    c.) find an actual root and use the quotient I found in part "b" to find the remaining roots and solve

    So here's what happened:
    1. I found all of the possible rational zeros
    2. I used synthetic division on each possible root, none of which equaled zero.
    3. For part "b" I assumed that there is no zero for this problem...
    4. I check the answer in the back of my textbook and found the answer to "b" :
    b.) -2
    and c.) {-2, 1 + √7, 1 - √7}

    I've done all I possibly can with what I know and I couldn't come up with the same answer in the textbook..
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: Can't figure out how.. (Pre-Calc, Rational Zero Theorem and Synthetic Division

    Since the leading coefficient is 1, the list of possible rational roots will be integral:

    \pm(1,2,3,4,6,12)

    Now, trying -2, we find:

    f(-2)=(-2)^3-10(-2)-12=-8+20-12=0

    So, we know x+2 is a factor. Performing the division, we find:

    \frac{x^3-10x-12}{x+2}=x^2-2x-6 and so:

    x^3-10x-12=(x+2)(x^2-2x-6)

    We know x=-2 is one root, the other two are found from the roots of the quadratic factor:

    x=\frac{2\pm2\sqrt{7}}{2}=1\pm\sqrt{7}
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    Re: Can't figure out how.. (Pre-Calc, Rational Zero Theorem and Synthetic Division

    Quote Originally Posted by MarkFL2 View Post

    Now, trying -2, we find:

    f(-2)=(-2)^3-10(-2)-12=-8+20-12=0

    So, we know x+2 is a factor. Performing the division, we find:

    \frac{x^3-10x-12}{x+2}=x^2-2x-6 and so:

    x^3-10x-12=(x+2)(x^2-2x-6)
    What theorem is that? I think I missed the lecture over that theorem...
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  4. #4
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    Re: Can't figure out how.. (Pre-Calc, Rational Zero Theorem and Synthetic Division

    I'm not sure if you saw my response, (I'm new at the moment, my apologies.)
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  5. #5
    MHF Contributor MarkFL's Avatar
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    Re: Can't figure out how.. (Pre-Calc, Rational Zero Theorem and Synthetic Division

    If r is a root of f(x), then f(r) = 0.
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    Re: Can't figure out how.. (Pre-Calc, Rational Zero Theorem and Synthetic Division

    Oh.... alright...
    That clears it up! Many thanks!
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