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Which of the following is a polynomial with roots

−3√3√−2


Ax^3 – 2x^2 – 3x + 6

Bx^3 + 2x^2 – 3x – 6

Cx^3 – 3x^2 – 5x + 15

Dx^3 + 3x^2 – 5x – 15

Verified Answer :

florarunyon

 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

dalendrk

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

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