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

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

Step-by-step explanation: Given roots of the polynomial
`-√(3) , √(3) , 2.`

Therefore, factors of the polynomial with given roots would be

`(x-√(3)), (x+√(3)), (x-2)`.

Multiplying those factors.

`(x-√(3))(x+√(3))(x-2)`.

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

`=x^2-3`

`\left(x-√(3)\right)\left(x+√(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.

Answer 2

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.