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17 December, 08:52

Write a polynomial in standard form axn+bxn-1+ ... given the following requirements.

Degree: 3, Zeros at (-1,0), (-5,0) and (-7,0) and y-intercept at (0,35). Leading coefficient is 1.

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  1. 17 December, 12:19
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    We are given zeros : (-1,0), (-5,0) and (-7,0).

    We can write those zeros as

    x=-1, x=-5 and x=-7.

    So, the factors of the polynomial would be

    (x+1), (x+5) and (x+7).

    We also given leading coefficent = 1.

    Let us multiply all those factors of the polynomila we got and then multiply by 1 finally.

    (x+1) * (x+5) * (x+7).

    Let us foil first two factors (x+1) * (x+5) first, we get

    x^2 + 5x + 1x + 5 = x^2 + 6x + 5.

    Therefore, (x+1) * (x+5) * (x+7) = (x^2 + 6x + 5) (x+7).

    Let us multiply (x^2 + 6x + 5) and (x+7), we get

    (x^2 + 6x + 5) * (x+7) = x^3 + 7x^2 + 6x^2 + 42x + 5x + 35.

    Combining like terms 7x^2 + 6x^2 = 13x^2 and 42x + 5x = 47x.

    Therefore,

    x^3 + 7x^2 + 6x^2 + 42x + 5x + 35

    = x^3 + 13x^2 + 47x + 35.

    If we multiply it by 1, we get same terms.

    And also if we plug x=0, we get

    (0) ^3 + 13 (0) ^2 + 47 (0) + 35 = 0+0+0+35 = 35.

    We can see that y-intercept is 35 there, because we get 35 on plugging x=0.

    Therefore, x^3 + 13x^2 + 47x + 35 is the polynomial in standard form that fulfil all the given requirements.
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