If a polynomial has one root in the form a+√b, it has a second root in the form of a_√b

When roots of polynomials occur in radical form, they occur as two conjugates.

That is,

The conjugate of (a + √b) is (a - √b) and vice versa.

To show that the given conjugates come from a polynomial, we should create the polynomial from the given factors.

The first factor is x - (a + √b).

The second factor is x - (a - √b).

The polynomial is

f (x) = [x - (a + √b) ]*[x - (a - √b) ]

= x² - x (a - √b) - x (a + √b) + (a + √b) (a - √b)

= x² - 2ax + x√b - x√b + a² - b

= x² - 2ax + a² - b

This is a quadratic polynomial, as expected.

If you solve the quadratic equation x² - 2ax + a² - b = 0 with the quadratic formula, it should yield the pair of conjugate radical roots.

x = (1/2) [ 2a + / - √ (4a² - 4 (a² - b) ]

= a + / - (1/2) * √ (4b)

= a + / - √b

x = a + √b, or x = a - √b, as expected.