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8 June, 04:18

Find a polynomial f (X) of degree 3 with real coefficients and the following zero 4, 2 - i

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  1. 8 June, 04:30
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    f (x) = x³ - 8x² + 21x - 20

    Step-by-step explanation:

    ∵ x = a + bi is a root of f (x)

    ∴ x = a - bi is the second root of f (x)

    ∴ f (x) = x² - (sum of the roots) x + (the product of the roots)

    ∵ 2 - i is a root of f (x)

    ∴ 2 + i is the other root of f (x) ⇒ (conjugate to each other)

    ∵ The sum of the roots = 2 - i + 2 + i = 4

    ∵ The product of them = (2 - i) (2 + i) = 4 + 2i - 2i - i² = 4 + 1 = 5⇒i² = - 1

    ∴ f (x) = x² - 4x + 5

    ∵ 4 is a root of f (x)

    ∴ x - 4 is a factor of f (x)

    ∴ f (x) = (x² - 4x + 5) (x - 4) = x³ - 4x² - 4x² + 16x + 5x - 20

    ∴ f (x) = x³ - 8x² + 21x - 20
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