Question

write the simplest polynomial function in standard form with the given roots.\\(\sqrt{3}\\) and 4\\(\bigcirc x^2 + 4x\sqrt{3} + 4\sqrt{3}\\)\\(\bigcirc x^3 - 4x^2 - 3x + 12\\)\\(\bigcirc x^2 - 4x\sqrt{3} + 4\sqrt{3}\\)\\(\bigcirc x^3 - 4x^2 - 3x - 36\\)

Explanation:

Step1: Set up factor form

If $r_1$ and $r_2$ are roots, the polynomial is $(x-r_1)(x-r_2)$. Here, $r_1=\sqrt{3}$, $r_2=4$, so:
$(x-\sqrt{3})(x-4)$

Step2: Expand using FOIL method

Multiply first, outer, inner, last terms:
$x\cdot x - x\cdot 4 - \sqrt{3}\cdot x + \sqrt{3}\cdot 4$

Step3: Simplify to standard form

Combine like terms and arrange:
$x^2 - (4+\sqrt{3})x + 4\sqrt{3}$

Step4: Check conjugate root requirement

For a polynomial with integer coefficients, if $\sqrt{3}$ is a root, $-\sqrt{3}$ must also be a root. The polynomial becomes $(x-\sqrt{3})(x+\sqrt{3})(x-4)$

Step5: Multiply first two factors

$(x^2 - (\sqrt{3})^2)(x-4)=(x^2-3)(x-4)$

Step6: Expand final polynomial

$x^2\cdot x - x^2\cdot 4 - 3\cdot x + 3\cdot 4$
Simplify: $x^3 - 4x^2 - 3x + 12$

Answer:

$\boldsymbol{x^3 - 4x^2 - 3x + 12}$