Question

Write the polynomial whose zeros are $2+\sqrt{3}$ and $2-\sqrt{3}$

• A. $x^2-2x+1$
• B. $x^2-4x+1$
• C. $x^2-7x+1$
• D. $x^2-4x-1$

Answer 1

The correct option is B $x^2-4x+1$

The zeros are $2+\sqrt{3}$ and $2-\sqrt{3}$
Let the quadratic polynomial be $(x-r_1)(x-r_2)$ where $r_1$ and $r_2$ are the zeros of the polynomial.
Thus the polynomial is
$[x-(2+\sqrt{3}\left)\right][x-(2-\sqrt{3}\left)\right]$
$=(x-2-\sqrt{3})(x-2+\sqrt{3})$
$=(x^2-2x+\sqrt{3}x-2x+4-2\sqrt{3}-\sqrt{3}x+2\sqrt{3}-3)$
$=x^2-4x+1$