Use the Rational Zero Theorem to find all real zeros

Question

8x³ − 15x² + 22x + 3 = 0

Answer 1

The rational root theorem states that the only possible rational roots of the cubic given here are of the form ± {1,3} / {1,2,4,8}.

Checking these individually, we find - 1/8 is a real root and we can use synthetic division to factor out (8x + 1):

-1/8 | 8. -15. 22 3

______-1___2___-3____

8 -16 24 0

8x³ - 15x² + 22x + 3 = (8x + 1) (x² - 2x + 3)

The remaining quadratic has no real roots, since its discriminant, b² - 4ac, is -8.