Question

Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
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Solution

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

straight alpha plus straight beta space equals space square root of 0 space space space rightwards double arrow space space minus straight b over straight a equals 0 space space rightwards double arrow space minus straight b space equals space 0

and

αβ space equals space square root of 5 space space space space rightwards double arrow space space straight c over straight a space equals space square root of 5 space space space rightwards double arrow space space straight c space equals space straight a. square root of 5

straight a equals 1 comma space space then space straight b space equals space 0 space and space straight c space equals space square root of 5.

So, quadratic polynomial which fits the given condition is

straight x squared plus 0. straight x plus square root of 5 space or space straight x squared plus square root of 5

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