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Polynomials

Question

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Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
$space 1 fourth comma space minus 1$

Solution

Let the quadratic polynomial be ax² + bx + c, and its zeroes be α and β. Then,

straight alpha plus straight beta equals 1 fourth space rightwards double arrow space space space fraction numerator negative straight b over denominator straight a end fraction equals 1 fourth space rightwards double arrow space space straight a space equals space minus 4 straight b

and

αβ space equals space minus 1 space space space rightwards double arrow space space space space straight c over straight a equals negative 1 space rightwards double arrow space straight c equals negative straight a

If

straight a space equals 1 comma space space then space straight b space equals space minus 1 fourth space space and space straight c space equals space minus 1.

So, quadratic polynomial which fits the given condition is

straight x squared minus 1 fourth straight x minus 1 space space space or space 4 straight x squared minus straight x minus 4.

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