Question

If 2+sqrt(3) is a polynomial root, name another root of the polynomial, and explain how you know it must also be a root

Answer 1

2 - √3

Explanation:

Find other root. Use x as a substitute.

`x = 2 + √(3)\\x - 2 = √(3)\\\left(x - 2\right)^2 = 3\\x^2 = 4 - 2 * x * 2 = 3\\x^2 - 4x + 1 = 0`

Now, we must use the quadratic formula to find the rest of the solution.

`x = (-b\pm √(b^2-4ac) )/(2a)\\x = (4\pm √(\left(-4\right)^2-4 * 1 * 1) )/(2)\\(4 \pm √(12) )/(2) = 2 \pm 3`

If one root of this polynomial [root] is 2 - √3, the other will be 2 + √3. Inversing that rule, the answer is 2 - √3.