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22 August, 12:37

Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1 and the given zeros for 3, 5 + square root of 5

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  1. 22 August, 13:04
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    The general form of a polynomial is: x² + bx + c = 0. Substituting the roots,

    3² + b (3) + c = 0

    3b + c = - 9 - - > eqn 1

    (5 + √5) ² + (5 + √5) (b) + c = 0

    (5+√5) b + c = - 30-10√5 - - > eqn 2

    Solving both equations simultaneously by subtracting eqn 2 from eqn 1,

    (-2 - √5) b = 21 + 10√5

    b = - 8 - √5

    Using eqn 1,

    3 (-8 - √5) + c = - 9

    -24 - 3√5 + c = - 9

    c = - 9 + 24 + 3√5

    c = 15 + 3√5

    Hence, the polynomial is: f (x) = x² + (-8 - √5) x + (15 + 3√5).
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