Question

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

A) x3 − 2x2 − 3x + 6
B) x3 − 3x2 − 5x + 15
C) x3 + 2x2 − 3x − 6
D) x3 + 3x2 − 5x − 15

Answer 1

C)

Explanation:

x3 + 2x2 − 3x − 6

Answer 2

Hello!

The answer is: C)
`x^(3)+2x^(2)-3x-6`

Why?

Factoring we have:

`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+√(3))(x-√(3))=0`

So,

`x_(1)=-2\\x_(2)=-√(3)\\x_(3)=√(3)`

Let's substitute the roots into the equation,

Substituting: -2

`(-2+2)(x+√(3))(x-√(3))=(0)(x+√(3))(x-√(3))=0`

Substituting:
`-√(3)`

`(x+2)*(-√(3)+√(3))*(x-√(3))=(x+2)*(0)*(x-√(3))=0`

Substituting:
`√(3)`

`(x+2)*(x+√(3))*(√(3)-√(3))=(x+2)*(x+√(3))*(0)=0`

Have a nice day!