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24 November, 10:31

Find a degree 3 polynomial with real coefficients having zeros 2 and 2 - 2 i and a lead coefficient of 1. Write P in expanded form. Be sure to write the full equation, including P (x) =.

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  1. 24 November, 10:43
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    The polynomial with real coefficients having zeros 2 and 2 - 2i is

    x³ - 6x² + 16x - 16 = 0

    Step-by-step explanation:

    Given that a polynomial has zeros at 2 and 2 - 2i, we want to write this polynomial.

    We have

    x - 2 = 0

    x - (2 - 2i) = 0

    => x - 2 + 2i = 0

    Since the polynomial has real coefficients, and 2 - 2i is a zero of the polynomial, the conjugate of 2 - 2i, which is 2 + 2i is also a polynomial.

    x - (2 + 2i) = 0

    => x - 2 - 2i = 0

    Now,

    P (x) = (x - 2) (x - 2 + 2i) (x - 2 - 2i) = 0

    = (x - 2) ((x - 2) ² - (2i) ²) = 0

    = (x - 2) (x² - 4x + 8) = 0

    = x³ - 4x² + 8x - 2x² + 8x - 16 = 0

    = x³ - 6x² + 16x - 16 = 0

    This is the polynomial required.
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