Which is the polynomial function of lowest degree with rational real coefficients, a leading coefficient of 3 and roots sqrt 5 and 2?

A) f (x) = 3x^3-6x^2-15x+30. B) f (x) = x^3-2x^2-5x+10. C) f (x) = 3x^2-21x+30. D) f (x) = x^2-7x+10.

The polynomial functions A) and C) have a leading coefficient of 3, but C) has a lower degree.

f (x) = 3 x² - 21 x + 30

3 (x² - 7 x + 10) = 3 (x² - 5 x - 2 x + 10) =

= 3 (x (x - 5) - 2 (x - 5)) = 3 (x - 2) (x - 5) = 0

x = 2 and x = 5 (roots)

Answer:

C) f (x) = 3 x² - 21 x + 30