Question

Find the polynomial of lowest degree with only real coefficients and having the zeros 3+sqrt(6),3-sqrt(6), and 1.

Answer 1

I know that the lowest possible degree of the polynomial is 3. The given zeros indicate that [x-(3+sqrt(6))],[x-(3-sqrt(6))], and (x-1) are factors.

Therefore f(x)=[x-(3+sqrt(6))][x-(3-sqrt(6))](x-1) and I need to multiply to find the polynomial. Thats where im stumped

Answer 2

A polynomial of lowest degree with zero(e)s a, b, and c, where a, b, c are distinct real numbers, is

f(x) = (x - a) (x - b) (x - c).

Here, a = 1 + √2, b = 1 - √2, c = 1. So

f(x)
= (x - 1 - √2) (x - 1 + √2) (x - 1)
= (x² - 2 x - 1) (x - 1)
= x³ - 3 x² + x + 1.

Answer 3

Thanks!