Questions: Find the coefficients of the function with roots, i sqrt(3) and a double root of 2.

Solution Steps

Since the roots are $i\sqrt{3}$, $-i\sqrt{3}$ and 2 (double root), the factors corresponding to these roots are $(x - i\sqrt{3})$, $(x + i\sqrt{3})$, $(x-2)$, and $(x-2)$.

Multiply the factors to obtain the polynomial: $(x - i\sqrt{3})(x + i\sqrt{3})(x-2)(x-2) = (x^2 + 3)(x^2 - 4x + 4) = x^4 - 4x^3 + 4x^2 + 3x^2 - 12x + 12 = x^4 - 4x^3 + 7x^2 - 12x + 12$.

The polynomial is $x^4 - 4x^3 + 7x^2 - 12x + 12 = 0$. Comparing this with the given form $x^4 + \diamond x^3 + \diamond x^2 - 12x + \diamond = 0$, we can identify the coefficients: Coefficient of $x^3$ is -4. Coefficient of $x^2$ is 7. Constant term is 12.

Final Answer