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Anonymous
Anonymous asked in Science & MathematicsMathematics · 2 months ago

What the zeros of the polynomial function ...?

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2 Answers

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  • 2 months ago

    The same logic as your last question is done here.  You need to find any rational roots so you can factor the polynomial down to a quadratic to find the last two roots.

    The Rational Root Theorem says that if there are any rational roots it will be from a list made up of all factors of the constant term over all factors of the high-degree polynomial.  In this case, that list is:

    ±1, ±2, ±4, ±8, ±16

    There are 10 possible rational roots that we'd have to brute-force to try to find one.

    Using the multiple-choice answers as a hint, we see it should be either -2 or +2.  So I'll test the twos and look for zeroes:

    f(x) = x³ + 5x² - 6x - 16

    f(-2) = (-2)³ + 5(-2)² - 6(-2) - 16 and f(2) = 2³ + 5(2)² - 6(2) - 16

    f(-2) = -8 + 5(4) + 12 - 16 and f(2) = 8 + 5(4) - 12 - 16

    f(-2) = -8 + 20 + 12 - 16 and f(2) = 8 + 20 - 12 - 16

    f(-2) = 8 and f(2) = 0

    We now know that x = 2 is a root which means that (x - 2) is a factor.  This means we can divide the cubic by this factor to get a quadratic quotient:

    . . . . _x²_+_7x_+_8___

    x - 2 ) x³ + 5x² - 6x - 16

    . . . . . x³ - 2x²

    . . . . -------------

    . . . . . . . . 7x² - 6x - 16

    . . . . . . . . 7x² - 14x

    . . . . . . . ---------------

    . . . . . . . . . . . . . 8x - 16

    . . . . . . . . . . . . . 8x - 16

    . . . . . . . . . . . . ------------

    . . . . . . . . . . . . . . . . . 0

    Now that we have the resulting quadratic to get the final two roots:

    x² + 7x + 8 = 0

    x = [ -b ± √(b² - 4ac)] / (2a)

    x = [ -7 ± √(7² - 4(1)(8))] / (2 * 1)

    x = [ -7 ± √(49 - 32)] / 2

    x = (-7 ± √17) / 2

    The three roots are:

    x = 2 and (-7 ± √17) / 2

    Answer D.

  • ?
    Lv 7
    2 months ago

     f(x) = x^3 + 5x^2 - 6x - 16

     f(x) = (x - 2) (x^2 + 7 x + 8)

     x = 2

     x = - 7/2 - sqrt(17)/2 

     x = - 7/2 + sqrt(17)/2 

     Answer choice:

     D

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