If c^4 - c + 1 = 0, then c^5 + c^(-5) = ?

Question

Please help. So far, I've only gotten to c(c - 1)(c^2 + c + 1) = -1 and can't find a way to convert the second equation in an useful form.

Answer 1

Hi Jeff! Since we want to find c^5 and c^(-5), let's try solving the first equation for c^4. We get c^4 = c - 1. To find c^5, we can multiply both sides of the equation by c. This gives us c^5 = c^2 - c. We can then use this new equation to find c^(-5). We can raise both sides of the equation to the (-1) power. We get (c^5)^(-1) = (c^2 - c)^(-1). We can simplify this to c^(-5) = 1 / (c^2 - c). Finally, we want to find c^5 + c^(-5), so let's add the two expressions. We get c^5 + c^(-5) = c^2 - c + 1 / (c^2 - c).