Question

For a given polynomial, a given zero is 2 + 5√3, what would be another zero? Ive tried to solve it myself but can't think of any ways to do it. If anyone could help, I'd appreciate it :)

Answer 1

2 - 5√3

Explanation:

This is because for a polynomial aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ where n ≥ 2, there would always be a polynomial of at least degree n = 2.

Now, for a polynomial of degree n = 2, a₂x² + a₁x + a₀, the roots of the equation are in conjugate pairs if the discriminant a₁² - 4a₂a₀ is not a perfect square. So, we would have a surd in the root and thus a conjugate pair.

So given a zero of 2 + 5√3, its conjugate is 2 - 5√3 which is the other zero.