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Math Help - Polynomial Express

  1. #1
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    Polynomial Express

    Solve:

    x^3-5x^2+8x-4 = 0

    x^3-5x^2+8x-4 > 0
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by nerdzor View Post
    Solve:

    x^3-5x^2+8x-4 = 0
    by the remainder theorem, we see that x = 1 is a root. thus by the factor theorem, (x - 1) is a factor.

    dividing our cubic by x - 1, we see that:

    x^2 - 5x^2 + 8x - 4 = (x - 1)(x^2 - 4x + 4) = (x - 1)(x - 2)^2 = 0

    thus, we have x = 1 or x = 2

    x^3-5x^2+8x-4 > 0
    i leave this part to you. we found our roots to be x = 1 and x = 2, so we in fact can split up the real line into the intervals (-\infty, 1), (1,2), and (2, \infty)

    on which of these intervals is the cubic positive?
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  3. #3
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    By inspection (x-1) is a factor because x = 1 is a zero of the expression

    then by use synthetic division

    x^3-5x^2+8x-4 = (x-1)(ax^2+bx+c)

    Compare coefficients of the highest order term to get a = 1 and lowest order to get c = 4 first because it is easy

    x^3-5x^2+8x-4 = (x-1)(x^2+bx+4) by comparing the coefficient of the term with x^2 or otherwise you should easily get b = -4

    x^3-5x^2+8x-4 = (x-1)(x^2-4x+4)
    \Rightarrow x^3-5x^2+8x-4 = (x-1)(x-2)^2

    for the inequality you must sketch a graph.

    You know be able to tell that for large positive values of x the function will be positive, and for large negative values of x the function will be negative.
    You should also know the the sign of the function can only change at a root.

    you should get a graph like this. the image below.
    Remember the inequality is greater than zero not greater than or equal to, don't be lazy and just write x >1
    Attached Thumbnails Attached Thumbnails Polynomial Express-picture-1.png  
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