Question

Find a quadratic polynomial whose zeroes are $(5-3\sqrt{2})$ and $(5+3\sqrt{2})$

Answer 1

If $5\pm 3\sqrt{2}$ are zeros of polynomial then

$x-(5+3\sqrt{2})$ and $x-(5-3\sqrt{2})$ are factors of the polynomial

$\therefore [x-(5+3\sqrt{2}\left)\right][x-(5-3\sqrt{2}\left)\right]$

$=x^2-(5-3\sqrt{2}+5+3\sqrt{2})x+(5+3\sqrt{2})(5-3\sqrt{2})$ [Since, $(x-a)(x-b)=x^2-(a+b)x+ab$]

$=x^2-x(5-3\sqrt{2}+5+3\sqrt{2})+(5^2-(3\sqrt{2}{)}^2)$

$=x^2-10x+25-9\left(2\right)$

$=x^2-10x+25-18$

$=x^2-10x+7$

Hence, $x^2-10x+7$ is required polynomial.