Which of the following is a polynomial with roots: −square root of 3, square root of 3, and −2?A) x3 − 2x2 − 3x + 6B) x3 − 3x2 − 5x + 15C) x3 + 2x2 − 3x − 6D) x3 + 3x2 − 5x − 15

Hello!The answer is: C) [tex]x^{3}+2x^{2}-3x-6[/tex]Why?Factoring we have:[tex]x^{3}+2x^{2}-3x-6=(x^{2})(x+2)-3(x+2)\\(x^{2})(x+2)-3(x+2)=(x+2)(x^{2}-3)\\(x+2)(x+\sqrt{3})(x-\sqrt{3})=0[/tex]So,[tex]x_{1}=-2\\x_{2}=-\sqrt{3}\\x_{3}=\sqrt{3}[/tex]Let's substitute the roots into the equation,Substituting: -2[tex](-2+2)(x+\sqrt{3})(x-\sqrt{3})=(0)(x+\sqrt{3})(x-\sqrt{3})=0[/tex]Substituting: [tex]-\sqrt{3}[/tex][tex](x+2)*(-\sqrt{3}+\sqrt{3})*(x-\sqrt{3})=(x+2)*(0)*(x-\sqrt{3})=0[/tex]Substituting: [tex]\sqrt{3}[/tex][tex](x+2)*(x+\sqrt{3})*(\sqrt{3}-\sqrt{3})=(x+2)*(x+\sqrt{3})*(0)=0[/tex]Have a nice day!