Question

Which of the following is a polynomial with roots negative square root of 3, square root of 3, and 2?

x3 − 2x2 − 3x + 6
x3 + 2x2 − 3x − 6
x3 − 3x2 − 5x + 15
x3 + 3x2 − 5x − 15

Answer 1

It's the first one. The way you figure that out is to turn each one of those roots into a factor of the polynomial, like this: if a root is 2, the factor is (x - 2). The factors of this polynomial are (x + sqrt(3)), (x - sqrt(3)), and (x - 2). When you multiply all of those together by FOILing, you get the first choice.

Answer 2

Answer:= $x^3-2x^2-3x+6.$

Step-by-step explanation: Given roots of the polynomial $-\sqrt{3} , \sqrt{3} , 2.$

Therefore, factors of the polynomial with given roots would be

$(x-\sqrt{3}), (x+\sqrt{3}), (x-2)$.

Multiplying those factors.

$(x-\sqrt{3})(x+\sqrt{3})(x-2)$.

$\mathrm{Expand}\:\left(x-\sqrt{3}\right)\left(x+\sqrt{3}\right)=x^2-\left(\sqrt{3}\right)^2$

$=x^2-3$

$\left(x-\sqrt{3}\right)\left(x+\sqrt{3}\right)\left(x-2\right)=\left(x^2-3\right)\left(x-2\right)$

$\mathrm{Expand}\:\left(x^2-3\right)\left(x-2\right)$

$=x^3-2x^2-3x+3\cdot \:2$

$=x^3-2x^2-3x+6$

Therefore, correct option is first option x^3-2x^2-3x+6.