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Thread: Polynomial Express

  1. #1
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    Polynomial Express

    Solve:

    $\displaystyle x^3-5x^2+8x-4 = 0$

    $\displaystyle x^3-5x^2+8x-4 > 0$
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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:

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

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

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

    thus, we have $\displaystyle x = 1$ or $\displaystyle x = 2$

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

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

    then by use synthetic division

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

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

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

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

    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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