Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
$square root of 2 comma space 1 third$

Solution

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

straight alpha plus straight beta equals square root of 2 space space space rightwards double arrow space minus straight b over straight a equals square root of 2 space space rightwards double arrow space minus straight b space equals space straight a. square root of 2

and

αβ equals 1 third space space rightwards double arrow space straight c over straight a equals 1 third space rightwards double arrow space straight a space equals space 3 straight c

straight a equals 1 comma space then space space straight b equals negative square root of 2 space and space straight c equals 1 third.

So, quadratic polynomial which fits the given condition is

straight x squared minus square root of 2 straight x plus 1 third space or space 3 straight x squared minus 3 square root of 2 straight x plus 1

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