Question

Find all six roots of the equation x^6 - 7x^3 - 8 = 0.

a) x = -2, x = 1, x = 2
b) x = -1, x = 2, x = -2
c) x = 1, x = -2, x = 2
d) x = -2, x = -1, x = 2

Answer 1

To find the roots of the equation x^6 - 7x^3 - 8 = 0, we can rewrite it as a quadratic equation u^2 - 7u - 8 = 0 by making a substitution. By factoring, we find the roots of u, which give us the six roots of x.

Step-by-step explanation:

The equation given is x^6 - 7x^3 - 8 = 0. To find the roots, we can rewrite the equation as (x^3) ^2 - 7x^3 - 8 = 0. Let's make a substitution, let u = x^3. Now our equation becomes u^2 - 7u - 8 = 0, which is a quadratic equation. We can solve this equation by factoring or using the quadratic formula.

By factoring, we can rewrite the equation as (u - 8) (u + 1) = 0. So, u = 8 or u = -1. Substituting u back as x^3, we get two sets of roots: x^3 = 8 or x^3 = -1. Taking the cube root, we find the six roots: x = 2, x = -2, x = 1.