Question

Find a quadratic polynomial with rational coefficients whose on zero is √5-2

Answer 1

Polynomial is x² + 4x - 1 = 0
Step by step explanation
i) The theory behind such problems are:
"In any real equation, the irrational as well imaginary roots occur only in pairs."
This is, if (a + √b) is one root, then the other root is (a - √b)
Similarly if (a + ib) is one root, then the other root is (a - ib)

ii) So here one root is given as: (-2 + √5), which is irrational.
So the other root of the polynomial is (-2 - √5)

iii) Thus in terms of roots, the polynomial is:
x² - (Sum of roots)x + (Product of roots) = 0

So here it is: x² - {(-2 + √5) + (-2 - √5)}*x + {(-2 + √5)*(-2 - √5)} = 0
= x² - (-4)x + (4-5) = 0
==> The polynomial as:x² + 4x - 1 = 0