Question

Given that $x-\sqrt{5}$ is a factor of the cubic polynomial $x^3-3\sqrt{5}{x}^2+13x-3\sqrt{5},$ Find all the zeros of the polynomial.

Answer 1

Given

polynomial is $x^3-3\sqrt{5}{x}^2+13x-3\sqrt{5}$

And $(x-\sqrt{5})$ is a factor.

So, we use long division to find the other factor as shown in the image,

$\therefore x^3-3\sqrt{5}{x}^2+13x-3\sqrt{5}=(x-\sqrt{5})(x^2-2\sqrt{5}x+3)$

Now using quadratic formula for $x^2-2\sqrt{5}x+3$ , We get

$x=\sqrt{5}\pm \sqrt{2}$

Hence the roots are $\sqrt{5},\sqrt{5}+\sqrt{2}$ and $\sqrt{5}-\sqrt{2}$