Rational root theorem

Question

Using the Rational Root Theorem, list all the possible rational roots of the polynomial.

f(x)=4x^5-7x^3+8x-18

Answer 1

For a polynomial of the standard form a_nxⁿ + a_n− 1x^n − 1 + … + a₁x¹ + a₀ = 0 (where a₀, a₁,…, a_n are integers) to have a rational solution p/q, q must divide a_n and p must divide a_0. That's the theory, let's look at your example:

a_n=4 and a₀=-18

What integers divide 4? I propose 1, 2, and 4 or -1, -2, and -4

And how about 18? 1, 2, 3, 6, 9, 18. And their negative values as well.

p can potentially be 1, 2, 3, 6, 9, 18, -1, -2, -3, -6, -9 and -18

q can potentially be 1, 2, 4, -1, -2, and -4

All the possible p/q's are: 1, 2, 3, 6, 9, 18, 1/2, 3/2, 9/2, 1/4, 3/4, 9/4, 18/4 and their negatives.