Show that √2 + √3 is irrational, by finding a monic polynomial of which it is the root.?

Answer 1

Let's let the roots be (√2 + √3) and (√2 - √3)

f(x) = (x - root1)(x - root2)
f(x) = (x - (√2 + √3))(x - (√2 - √3))
f(x) = x² - (√2 - √3)x - (√2 + √3)x + (√2 + √3)(√2 - √3)
f(x) = x² -(√2)x + (√3)x - (√2)x + -(√3)x + (2 + 3)
f(x) = x² -2(√2)x + 5