Question

Which of the following is a polynomial with roots

−3√3√−2


A. x^3 – 2x^2 – 3x + 6

B. x^3 + 2x^2 – 3x – 6

C. x^3 – 3x^2 – 5x + 15

D. x^3 + 3x^2 – 5x – 15

Answer 1

the polynomial has the factor (x-n)
So any polynomial with these roots, must have the factors (x+sqrt(3))(x-sqrt(3))(x+2)
Since these are cubic polynomials (highest x term is x^3), these are all the factors, so the polynomial is of the form:
k(x+sqrt(3))(x-sqrt(3))(x+2)
And since the x^3 term is 1, the constant k must be 1.

So, the polynomial must be the one which is equal to:
(x+sqrt(3))(x-sqrt(3))(x+2)
Since the constant term (the last term) is different for each of your 4 options, you just need to evaluate the constant term and see which one matches.
i.e. the answer is whichever polynomial has a constant term equal to
sqrt(3) times (-sqrt(3)) times 2

Answer 2

`a(x-x_1)(x-x_2)(x-x_3)\\\\x_1=-\sqrt3;\ x_2=\sqrt3;\ x_3=-2;\ a=1\\\\subtitute\\\\(x+\sqrt3)(x-\sqrt3)(x+2)=(x^2-3)(x+2)\\\\=\boxed{x^3+2x^2-3x-6}\to\fbox{B.}`