Question

Find a quadratic polynomial whose zeros are $3 \pm \sqrt 2 $ .

Answer 1

Hint: In such questions we should know how to express a quadratic polynomial in the form of the sum of its zeroes and product of its zeroes to reach the solution of the problem . Use the formula of ${a^2} - {b^2}$ = ( a+b )( a-b ) to simplify the calculation .

Complete step-by-step answer:

Given are the zeroes of the quadratic polynomial $3 + \sqrt 2 $ and $3 - \sqrt 2 $
We know that a quadratic polynomial = ${x^2}$ - ( sum of zeroes ) x + ( product of zeroes )
Sum of zeroes = $\left( {3 + \sqrt 2 } \right) + \left( {3 - \sqrt 2 } \right) = 6$
Product of zeroes = $\left( {3 + \sqrt 2 } \right)\left( {3 - \sqrt 2 } \right)$
$ = {3^2} - {\sqrt 2 ^2} = 9 - 2 = 7$
Hence the quadratic polynomial is ${x^2} - 6x + 4$ .

Note –
In such types of questions the key concept is to remember all the expressions of a quadratic polynomial along with the basic formulas used for calculation to get to the desired result in a faster and accurate way .