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23 May, 08:47

Write a polynomial equation with roots 5 and - 9i. X^3-? x^2+? X-?=0

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  1. 23 May, 11:35
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    x³ - 5x² + 81x - 405 = 0

    Step-by-step explanation:

    Complex roots occur in conjugate pairs.

    Thus given x = - 9i is a root then x = 9i is also a root

    The factors are then (x - 5), (x - 9i) and (x + 9i)

    The polynomial is the the product of the roots, that is

    f (x) = (x - 5) (x - 9i) (x + 9i) ← expand the complex factors

    = (x - 5) (x² - 81i²) → note i² = - 1

    = (x - 5) (x² + 81) ← distribute

    = x³ + 81x - 5x² - 405, thus

    x³ - 5x² + 81x - 405 = 0 ← is the polynomial equation
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