Use the rational root theorem to list all possible rational roots for the equation. x³+ 2x - 9 = 0.

Solution:

Using the Rational Roots(Zeros) Theorem, which states that, if the polynomial f(x) = a_nxⁿ + a_n−1x^n-1 +...+ a₁x + a₀ has integer coefficients, then every rational zero of f(x) has the form p/q where p is a factor of the constant term a₀ and q is a factor of the leading coefficient aₙ.

Given f(x) = x³+ 2x - 9

Here,

p: ±1, ±3, ±9 which are all factors of constant term 9

q: ±1 which are the factors of the leading coefficient 1

All possible values are

p/q: ±1, ±3, ±9

Given, f(x) = x³+ 2x - 9

f(1) = 1 + 2 - 9 = -6

f(-1) = -1 -2 -9 = -12

f(3) = 27 + 6 -9 = 24

f(-3) = -27-6 -9 = -57

f(9) = 729 + 18 - 9 = 738

f(-9) = -729 - 18 - 9 = -756

No values are leading to zeros hence, none of these can be roots of the equation.