Question

which of the following is a polynomial with roots -√(3),√(3),2

Answer 1

You didn't give me any choices, but no matter. Here's what you do:

a polynomial \(f(x)\) with roots \(r_1, r_2,...r_n\) can be written in the form
\[f(x) = k(x-r_1)(x-r_2)...(x-r_n)\]
Plug in your roots, multiply it out, and you've got your polynomial. \(k\) is a scaling constant used to make the polynomial pass through an arbitrary point; a moment's thought ought to tell you that the value of \(k\) does not affect the roots. If it doesn't tell you that, solve the following equations:

\[x^2 = 0\]\[2x^2 = 0\]

Answer 2

If it isn't perfectly clear, you can choose any value of \(k\) you like (probably \(k = 1\)) if they aren't asking you to make the curve go through a specific point. If it does have to go through a specific point \((x_0,y_0)\), then you plug in \(x = x_0\), and solve for the value of \(k\) that makes \(y = y_0\).