Question

Quadratic polynomials with rational coefficients having $\sqrt{3}$ as a zero are given by ________.

Answer 1

Irrational zeroes of a quadratic polynomial always occurs in pairs.
If one zero is $\sqrt{3}$ then, other zero is $-\sqrt{3}$ −3-\sqrt 3

Now,
Quadratic polynomial with given zeroes is
x² – (sum of zeroes)x + (product of zeroes)

=x2−(3−3)x+(3×−3)=x^2 - (\sqrt3 -\sqrt 3)x+(\sqrt3\times-\sqrt3)

$=x^2-(\sqrt{3}-\sqrt{3})x+(\sqrt{3}\times -\sqrt{3})$

$=x^2-3$

Hence, Quadratic polynomials with rational coefficients having 3 as a zero are given by x2−9x^2-9x2−3.