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  1. #1
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    Algebra Question: How do you factor this?

    In my Algebra 2 class we had a problem that not even the teacher knew the method that was used to factor down the problem.

    Here's the final step before the factoring;

    0 = X3 – 18x2 + 107x - 216

    Now this is what the book said the answer was;

    0 = X3 – 18x2 + 107x - 216
    0 = (X-8)(X2-10x+27)
    0 = X-8
    X = 8

    So, could anyone help as to figuring out what method was used, and how the problem was factored down?

    And yes, I couldn't make any of the text superscript, so I made the exponents smaller and red.

  2. #2
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    looking through my notes...

    Isn't it just grouping? I'll solve it, but it looks like regular grouping.
    Nope... I'll work on it


    TO THE POST UNDER ME: Doesnt fit into the quadratic equation.
    Last edited by Xshare; 02-18-2004 at 07:50 PM.
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  3. #3
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    Nope, I was wrong I'll have a look at it properly now!

  4. #4
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    You would have to substiture X with numbers so you equation would be equal to 0. In this case, 8 is the number, when you plug in 8 in place of X, that would wind up to 8^3 - 18(8)^2 + 107(8) - 216 = 0.

    Then you divide the given equation: X3 – 18x2 + 107x - 216
    by (x-8), which would lead to X2-10x+27.

    Hope that helps, If you need more explantion, I would be more than glad to help you out.

  5. #5
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    Del, that method is only good if you know the answer

  6. #6
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    Yeah, that's just checking to make sure that the answer is correct. I need to figure out how they got that answer.

  7. #7
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    Originally posted by Joseph_M
    Del, that method is only good if you know the answer
    Well, it can be done with easy numbers. However, you cannot say that it;s only good if I know the answer.

  8. #8
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    Hmm.. well 107 is prime. I'm always hitting that stumbling block.
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  9. #9
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    Easy numbers, we're talking it can be done reasonably quickly up to 4 D.Ps, but its time-consuming for anything over that, there is a formula to work it out with, I just don't have it to hand, I'll ask my mathematician friend, for he shall surely know the formula, or will have it on a piece of paper at his place.

  10. #10
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    You're still using the FOIL method except for a longer equation...

    X3 – 18x2 + 107x - 216

    0 = (x - 8) (x2 - 10x + 27)

    F: x * x2 = x3

    O: x * 27 = 27x
    O: x * -10x = -10x2

    I: -8 * 10x = 80x
    I: -8 * x2 = -8x2

    (Add your outer/inners together to get +107x and -18x2)

    L: 8 * 27 = 216

    Put it back together and you get...

    X3 – 18x2 + 107x - 216

  11. #11
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    Thats still only knowing the answer.. we're talking without knowing the answer.
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  12. #12
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    (x^3-y^3)=(x-y)(x^2+xy+y^2)
    (x+y)^3=x^3+3x^2y+3x^2y+y^3
    (x-y)^3=x^3-3x^2y+3x^2y-y^3

    you don't use that much foiling when you get to pre-calc/calc. just understand the major concepts.

    after this ask the teacher this:

    if (x)^2=(y)^2, then x=y

    (1)^2=2
    (-1)^2=2
    2=2, therefore (1)^2=(-1)^2), therefore ...

    1=-1?

    good luck.

  13. #13
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    Umm.. what y?
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  14. #14
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    0 = X3 – 18x2 + 107x - 216

    Factor out one x
    x (x2-18x+107-216)
    Combine like terms
    x (x2-18x-109)
    Take half of the middle term, then square it, add it in the middle, then subtract it on the end
    x (x2-18x+81) -81-109
    Ask, what multiples to get 81 and adds to get -18?
    x (x-9)(x-9) - 190
    190 = x(x-9)(x-9)

    Hmm... that didn't work... maybe this is a synthetic division problem... or maybe you should graph it and just find the zeros...
    Last edited by brandonk; 02-18-2004 at 08:39 PM.

  15. #15
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    Originally posted by ToaD

    2=2, therefore (1)^2=(-1)^2), therefore ...

    1=-1?

    good luck.
    I think you made a typo.... -1^2 is 1. And 1^2 isn't two, neither is -1^2

    TO BRANDON:
    No, you messed up. You can't factor an X out of 216. I've found out how you get the answer, I just need to work out the specifics.
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  16. #16
    Use synthetic prositution oh wait I mean subsitution, it works every time.

  17. #17
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    Originally posted by brandonk
    x (x2-18x+107-216)
    I don't think that would work for 216..

  18. #18
    using synthetic subsititution, sure it would, you just have to know how to use it.

  19. #19
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    Basically, you need to factor out 18x2 + 107x - 216 first. I know theres a solution, i just am too lazy to find it.
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  20. #20
    Originally posted by ToaD

    if (x)^2=(y)^2, then x=y
    if x^2=y^2,
    x does not necessarily equal y, as you've shown in your example. if x is -1, and y is 1, then x^2=y^2=1. This proves either that you either missed the logic, or gave us an absurdly easy brainteaser.

    It seems that the superscript tag for vBcode has been disabled. Any particular reason for that?

  21. #21
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    Originally posted by Xshare
    Basically, you need to factor out 18x2 + 107x - 216 first. I know theres a solution, i just am too lazy to find it.
    Of course there is a solution... it's just not easily found... it would require a lot of guessing to figure out what zeros to use for synthetic division, then use syn div to confirm your answer. My bet is that you would need to graph this equation on a graphing calculator... unless of course you forumlated it.

    Send a complaint to the author of the book damn it! They charge so much for those books, they should be perfect!

  22. #22
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    Nevermind, you can't it's unfactorable.
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  23. #23
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    Originally posted by Reuven
    if x^2=y^2,
    x does not necessarily equal y, as you've shown in your example. if x is -1, and y is 1, then x^2=y^2=1. This proves either that you either missed the logic, or gave us an absurdly easy brainteaser.

    It seems that the superscript tag for vBcode has been disabled. Any particular reason for that?
    I think he got the rule a little backwards...

    2^x=2^y then x=y

    If the bases are the same, then the exponents drop down and equal each other... but the bases don't equal each other

  24. #24
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    Re: Algebra Question: How do you factor this?

    Originally posted by inogenius
    0 = X3 – 18x2 + 107x - 216
    0 = (X-8)(X2-10x+27)
    0 = X-8
    X = 8
    Assuming you are working with real numbers (no imaginary numbers involved)

    Let f(x)=x^3 - 18x^2 + 107x - 216

    You need to realize that the root of f(x) will make f(x)=0. Thus you can plot the graph on your graphing calculator and see at what x values does f(x) = 0. Say, f(x) = 0 at three values, x=a, b, c then factors of f(x) = (x-a)(x-b)(x-c).

    Alternatively, if you do not have graphing calculator or if it's not allowed, then you can try plugging numbers into the original equation f(x), say x=1. Does f(1) = 0? If not, then (x-1) is not a factor of f(x). Try x=2. Does f(2) = 0? If not, then (x-2) is not a factor of f(x). You will see that x=8 will make f(x) = 0. Thus (x-8) is one of the factor of f(x). After this, find g(x)=f(x)/(x-a). Then find a factor of g(x) with the same method (find x such that g(x)=0).

    But once g(x) has been reduced to a polynomial of degree 2, then it's very easy to find its remaing roots/factors of f(x).
    Last edited by FHDave; 02-18-2004 at 09:43 PM.
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  25. #25
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    Originally posted by brandonk
    0 = X3 – 18x2 + 107x - 216

    Factor out one x
    x (x2-18x+107-216)
    Do you see "216x" in the original equation?
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