 QUESTION

# Find the quadratic polynomial if the sum and product of its zeroes are $\sqrt 3 , - 1$  A. ${x^2} - \sqrt 3 x + 1$ B. ${x^2} + \sqrt 3 x - 1$ C. ${x^2} - \sqrt 3 x - 1$ D. ${x^2} + \sqrt 3 x + 1$

Hint- There is a direct relation between the formation of quadratic polynomial equations and the roots. Roots are also called zeroes. The equation can be formed with the help of the sum of its roots and product of its roots.

Complete step-by-step solution -
In the question, we are given with the sum of zeroes as $\sqrt 3$ and the product of its zeroes as$- 1$.
The general form of any quadratic polynomial equation in term of x is
${x^2} - ({\text{sum of roots)}}x + ({\text{product of roots)}}$
So, by directly assigning the values at the required positions, we will obtain our quadratic polynomial equation.
So, the sum of roots is $\sqrt 3$
And product of roots is $- 1$
After assigning the values, our equation will become as
${x^2} - (\sqrt 3 )x + ( - 1)$
After simplifying the equation, our equation becomes
${x^2} - \sqrt 3 x - 1$
This is the same as option C.
So, C is our correct option.

Note-There is always a relation between the polynomial equation and their roots. Like in the given question, we only need to determine the correct relation between the formation of equation and the roots of equation. There we have relations between the roots and the coefficients of ${x}^{2}$ , $x$ and constant (c) for any quadratic polynomials or equations.