Question

3. If one zero of the polynomial x2 − 4x + 1 is 2 + √3. Write the other zero​

Answer 1

Given,

A polynomial $x^{2} -4x+1$ with one of its zero $2+\sqrt{3}$.

To find,

Other zero of the given polynomial.

Solution,

For a polynomial of the form $a^{2}+bx+c$ having roots $\alpha$ and $\beta$, we can establish a relation between sum of roots and product of roots with the coefficients of the polynomial.

We have,

• Sum of roots $(\alpha + \beta) = -b/a$

• Product of roots  $(\alpha\beta) = c/a$

Comparing the given quadratic equation with the general equation:

    $a = 1\\b = -4\\c = 1$

Let, $\alpha = 2+\sqrt{3}$ as one of its zero. (Given)

We have the relation, $(\alpha + \beta) = -b/a$

Substituting the values,

              $2 + \sqrt{3} + \beta = 4\\\beta + \sqrt{3} = 2 \\\beta = 2 - \sqrt{3}$

Therefore, the other zero of the polynomial is $2 - \sqrt{3}$